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COMPRESSED  AIR 


THEORY   AND   COMPUTATIONS 


BY 

ELMO    G.   HARRIS,   C.E. 

PROFESSOR  OF  CIVIL  ENGINEERING,    MISSOURI  SCHOOL  OF  MINES,, 

IN  CHARGE    OF  COMPRESSED  AIR  AND  HYDRAULICS; 

MEMBER    OF    AMERICAN    SOCIETY    o  ' 

CIVIL    ENGINEERS 


FIRST  EDITION 
SECOND  IMPRESSION  CORRECTED 


McGRAW-HILL   BOOK   COMPANY 

239  WEST   89TH  STREET,    NEW   YORK 

6  BOUVERIE  STREET,  LONDON,   E.G. 

1910 


Engineering 
Library 


MECHANICS   DEPT. 

COPYRIGHT,  1910, 

BY  THE 

McGRAW-HILL  BOOK  COMPANY 


Stanbopc  ipress 

F.    H.   CILSON     COMPANY 
BOSTON,     U.S.A. 


PREFACE* 


THIS  volume  is  designed ,  to  present  the  mathematical 
treatment  of  the  problems  in  the  production  and  applica- 
tion of  compressed  air. 

It  is  the  author's  opinion  that  prerequisite  to  a  successful 
study  of  compressed  air  is  a  thorough  training  in  mathe- 
matics, including  calculus,  and  the  mathematical  sciences, 
such  as  physics,  mechanics,  hydraulics  and  thermodynamics. 

Therefore  no  attempt  has  been  made  to  adapt  this  volume 
to  the  use  of  the  self-made  mechanic  except  in  the  insertion 
of  more  complete  tables  than  usually  accompany  such  work. 
Many  phases  of  the  subject  are  elusive  and  difficult  to  see 
clearly  even  by  the  thoroughly  trained;  and  serious  blunders 
are  liable  to  occur  when  an  installation  is  designed  by  one 
not  well  versed  in  the  technicalities  of  the  subject. 

As  one  advocating  the  increased  application  of  compressed 
air  and  the  more  efficient  use  where  at  present  applied,  the 
author  has  prepared  this  volume  for  college-bred  men,  believ- 
ing that  such  only,  and  only  the  best  of  such,  should  be 
entrusted  with  the  designing  of  compressed-air  installations. 

The  author  claims  originality  in  the  matter  in,  and  the  use 
of,  Tables  I,  II,  III,  V,  VI,  VII  and  IX,  in  the  chapter  on 
friction  in  air  pipes  and  in  the  chapter  on  the  air-lift  pump. 

Special  effort  has  been  made  to  give  examples  of  a  practical 
nature  illustrating  some  important  points  in  the  use  of  air 
or  bringing  out  some  principles  or  facts  not  usually  appre- 
ciated. 

Acknowledgment  is  herewith  made  to  Mr.  E.  P.  Seaver 
for  tables  of  Common  Logarithms  of  Numbers  taken  from 
his  Handbook. 

735781 


CONTENTS 

PAGE 
SYMBOLS  AND  FORMULAS ix,  xl 

CHAPTER  I. 

Art.    1.   Formulas  for  Work.  —  Temperature  Constant. . .  1 

Art.    2.   Formula  for  Work.  —  Temperature  Varying. ...  3 

Art.    3.   Formula  for  Work.  —  Incomplete  Expansion ...  7 

Art.    4.   Effect  of  Clearance.  —  In  Compression 8 

Art.    5.   Effect  of  Clearance  and  Compression  in  Expan- 
sion Engines 9 

Art.    6.   Effect  of  Heating  Air  as  it  Enters  Cylinders  ....  11 
Art.    7.   Change  of  Temperature  in  Compression  or  Ex- 
pansion    12 

Art.    8.   Density  at  Giyen  Temperature  and  Pressure. ...  13 

Art.    9.   Cooling  Water  Required 14 

Art.  10.   Reheating  and  Cooling 14 

Art.  11.   Compounding 16 

Art.  12.    Proportions  for  Compounding 19 

Art.  13.    Work  in  Compound  Compression 20 

Art.  14.   Work  under  Variable  Intake  Pressure 21 

Art.  15.   Exhaust  Pumps 22 

Art.  16.   Efficiency  when  Air  is  Used  without  Expansion.  .  24 

Art.  17.   Variation  of  Free  Air  Pressure  with  Altitude.  .  .  25 

CHAPTER  II.     MEASUREMENT  OF  Am. 

Art.  18.   General  Discussion 27 

Art.  19.   Apparatus  for  Measuring  Air  by  Orifice 28 

Art.  20.   Formula  for  Standard  Orifice  under  Low  Pressure  29 

Art.  21.   Air  Measurement  in  Tanks 30 

CHAPTER   III.     FRICTION  IN  AIR  PIPES. 

Art.  22.    General  Discussion 33 

Art.  23.   Friction  Formula  assuming  Density  and  Tem- 
perature Constant 33 

Art.  24.   Theoretically  Correct  Friction  Formula 36 

Art.  25.   Efficiency  of  Power  Transmission  by  Compressed 

Air..' 39 

CHAPTER  IV.     HYDRAULIC  AND  CENTRIFUGAL  AIR  COMPRES- 
SORS. 

Art.  26.    Displacement  Type  of  Air  Compressor 41 

Art.  27.    Entanglement  Type  of  Air  Compressor 42 

Art.  28.   Centrifugal  Type  of  Air  Compressor 44 

vii 


Viii  CONTENTS 

PAGB 
CHAPTER  V.     SPECIAL  APPLICATIONS  OF  COMPRESSED  AIB. 

Art.  29.   The  Return-Air  System  —  In  General 45 

Art.  30.   The  Return-Air  Pumping  System 46 

Art.  31.   The  Simple  Displacement  Pump 48 

CHAPTER  VI.   THE  AIR-LIFT  PUMP. 

Art.  32.   General  Discussion 49 

Art.  33.   Theory  of  the  Air-Lift  Pump 49 

Art.  34.    Design  of  Air-Lift  Pumps 52 

Art.  35.   The  Air  Lift  as  a  Dredge  Pump 57 

Art.  36.   Testing  Wells  with  the  Air  Lift 57 

Art.  37.   Data  on  Operating  Air  Lifts 58 

CHAPTER  VII.     EXAMPLES  AND  EXERCISES 

Art.  38 60 

PLATES  AND  TABLES 71 

APPENDIX  A.    Drill  Capacity  Tables 90 

APPENDIX  B.    Data  on  Friction  in  Air  Pipes 100 

APPENDIX  C.    Methods  and  Results  of  Experiments  at  Missouri 
School  of  Mines  Determining  Friction  in  Pipes 

and  Fittings 122 


SYMBOLS 


For  ready  reference  most  of  the  symbols  used  in  the  text  are  assembled 
and  defined  here. 

p  =  intensity  of  pressure  (absolute),  usually  in  pounds  per  square 
foot.  Compressed-air  formulas  are  much  simplified  by 
using  pressures  measured  from  the  absolute  zero.  Hence 
where  ordinary  gage  pressures  are  given,  p  =  gage  pres- 
sure +  atmospheric  pressure.  In  the  majority  of  formulas 
p  must  be  in  pounds  per  square  foot,  while  gage  pressures  are 
given  in  pounds  per  square  inch.  Then  p  =  (gage  pressure  + 
atmospheric  pressure  in  pounds  per  square  inch)  X  144. 

v  =  volume  —  usually  in  cubic  feet. 

Where  sub-a  is  used,  thus  pa,  va,  the  symbol  refers  to  free 
air  conditions. 

.  .  .  higher  pressure 

r  =  ratio  of  compression  or  expansion  =  f-^    -* 

lower  pressure 

The  lower  pressure  is  not  necessarily  that  of  the  atmos- 
phere. 

t  =  absolute  temperature  =  Temp.  F.  +  460.6. 
n  =  an  empirical  exponent  varying  from  1  to  1.41. 
loge  =  hyperbolic  logarithm  =  (common  log.)  X  2.306. 
W  =  work  —  usually  in  foot-pounds  per  second. 
Q  =  weight  of  air  passed  in  unit  time. 
w  =  weight  of  a  cubic  unit  of  air. 

Other  symbols  are  explained  where  used. 


FORMULAS 


For  convenience  of  reference  the  principal  formulas  appearing  in  the 
text  are  collected  here  with  the  article  and  page  where  demonstration 
and  complete  explanation  can  be  found. 

No.  Formula  Art.     Page 

1.  W  =  pv\0ger  ..........  ........................          1  2 

la.    W  =  53.17  t  loger  for  one  pound  .................       1          2 

lb.    W  =  (122.61  logior)  t  for  one  pound  ................       1          2 

2.  W  =  63737  logior  for  one  pound  at  60°  F  ...........       1          2 

logio  63737  =  4.8043894. 


4.  pv  =  53.17  1  for  one  pound  ........................       2         3 

5.  pivin  =  p-2v2n  ..........................  ..........       2          3 


6.      W  - 


Ti   —   1 

7- 
8 


|-n-i  -. 

.      W  =  ^-j-   piz>i  [r  "    -  ij 


8b.    W  =  95190  (r°-29  -  1)  for  1  lb.  at  60°  F.,  n  =  1.41  ...       2          5 
8c.    W  =  138405  (r°-2  -  1  )  for  1  lb.  at  60°  F.,  n  =  1.25  ...       2          5 
logio  95190  =  4.978606,  log  138405  =  5.141141. 


8d 


.    W  =  [53.17  ^—  (r~^~  -  l)J  t  for  one  pound  ......       2          5 


9.      W  =  —     ^-TJ —  +  ^2^2  —  paVi  for  partial  expansion  3  8 

10.      Ev  =  1  +  c  (1  —  r)  volumetric  efficiency 4  10 

lOa.      *  "Ml"! 5  u 

i-i 


11.       U  = 

lla.      fe  =  M^i»      =  'iW   n    •  7         13 


xii  FORMULAS 

No.                          Formula  Art.  Page 

12.       w  —  _Q??-    =  weight  per  cubic  foot  ...............  8  13 

53.17  1 

12a.     w  =  2.708  (jj^jpJTT?)  =  wei8ht  Per  cubic  foot  •  •  •  •  8  13 

12b.     dz=-^=\     d3=  —  ;  diameters,  stage  compression..  12  19 


(n  —  i         \ 
n~"~  -  1)  X  2;  two-stage  work  ----      13        20 
n  —  i  " 

/  n~l       \ 
13a.    W  =^-iPaVaV22n  -1JX2;    two-stage  work....     13        20 

/   n~l       \ 
13b.    W  =  ^-^PaVa  Vn  »    -VX  3;  three-stage  work.  .  .     13         21 

n  i   ILll       \ 

13c.    W  =  ^—  j-poVaUa  3  B  -  VX  3;  three-stage  work..  .     13         21 


15         23 


.           16  24 

rlo&r 

16.  pa=.4931m{l  -  .0001  (F  -32)] .../. 17  25 

17.  log  pa  =  1.16866  -12^47 17  26 

18.  WeightQ  =  c  .1632^ \  \Pa]  Pa  in  Ibs.  per  sq.  in 20  29 

18a.                 Q  =  c  .6299  d2  V  *-  at  sea  level 20  30 

20.  /  =  4^T! 23  35 

21.  d-(c}^)1 23  35 

24.      log^log^-C 24  37 

«  39 


26.       *-~^j 26  42 

27         Va   -       l      n        d  QQ  w 

**  •         /^     —    -7.7  Q    TT   ~j          ~~ oo  Oo 

y         77.O   h     Iogi0r 

28.       sx  =  va[l  -XA\  -  Ml  33  56 


COMP&SSSED   A 


CHAPTER  S'iV" 
FORMULAS  FOR  WORK 

Art.  1.  Temperature  Constant  or  Isothermal  Conditions. 

From  the  laws  of  physics  (Boyle's  Law)  we  know  that 
while  the  temperature  remains  unchanged  the  product  pv 
remains  constant  for  a  fixed  amount  (weight)  of  air.  Hence 
to  determine  the  work  done  on  or  by  air  confined  in  a  cylinder, 
or  like  conditions,  when  conditions  are  changed  from  p\v\  to 
P2V2  we  can  write 

p&i  =  pxvx  =  PzVz, 

sub  x  indicating  variable  intermediate  conditions. 


L 

p- 

i 
i 
i 

*-*» 

L/- 
3 

o 

Lr 

-*T- 

i 

Fig.  1. 
Whence  px  =  ^^  and  dW  =  pxAdl  =  pxdvx  since  Adi  =  dv; 

Vx 

being  the  area  of  cylinder,  therefore  dW  =  p&i  — ,  and 

Vx 

1 


2  COMPRESSED  AIR 

work  of  compression  or  expansion  between  points  B  and  C 
(Fig.  1)  is  the  integral  of  this,  or 

W  =  PiVi    I     '  —  =  piVi  (loge  Vi  —  loge  02) 

Jv2      Vx 

=  pivi\oger  =  p2v2\oger. 


Note  that  this  analysis  i$  only  for  thp.work  against  the/ron£ 
of  the  pisto^VJijl'et^gsjjigJfj'pna^  t(xcC.  To  get  the  work 
done  during  the  entire  stroke  of  piston  from  B  to  D  we  must 
note  that  throughout  the  stroke  (in  case  of  ordinary  compres- 
sion) air  is  entering  behind  the  piston  and  following  it  up 
with  pressure  p\.  Note  also  that  after  the  piston  reaches  C 
(at  which  time  valve  /  opens)  the  pressure  in  front  is  constant 
and  =  p2  for  the  remainder  of  the  stroke.  Hence  the  com- 
plete expression  for  work  done  by,  or  against,  the  piston  is 

loge  r  —  piVi  + 


but  since  p&i  =  p2v2,  the  whole  work  done  is 

W  =  piVi  loge  r    or     p2v2  loge  r.  (1) 


Note  that  the  operation  may  be  reversed  and  the  work 
be  done  by  the  air  against  the  piston,  as  in  a  compressed-air 
engine,  without  in  any  way  affecting  the  formula. 

Forestalling  Art.  2,  Eq.  (4),  we  may  substitute  for  pv  in 
Eq.  (1)  its  equivalent,  53.17  t,  for  one  pound  of  air  and  get 
for  one  pound 

W  =  53.17  ZX  loge  r.  (la) 

This  may  be  adopted  for  common  logs  by  multiplying  by 
2.3026.     It  then  becomes 

W  =  (122.61  Iog10  r)  t,  (Ib) 

(log  122.61  =  2.0878852.) 

Note  that  in  solving  by  logs  the  log  of  log  r  must  be  taken. 
Values  of  the  parenthesis  in  Eq.  (Ib)  are  given  in  Table  I 

For  the  special  temperature  of  60°  F.  (Ib)  becomes  for  one 
pound  of  air 

W  =  63737  log,0r,  (2) 

log  63737  =  4.8043894. 


FORMULAS  FOR  WORK  3 

Example  la.  What  will  be  the  work  in  foot-pounds  per 
stroke  done  by  an  air  compressor  displacing  2  cubic  feet  per 
stroke,  compressing  from  pa  =  14  Ibs.  per  sq.  inch  to  a  gage 
pressure  =  70  Ibs.;  compression  isothermal,  T  =  60°  F.? 

Solution  (a): 

Inserting  the  specified  numerals  in  Eq.  (1)  it  becomes 

W  =  144  X  14  X  2  X  lo&—  jt  U  =  4032  X  L79  =  7217- 

Solution  (b):  By  Tables  I  and  II. 

By  Table  II  the  weight  of  a  cubic  foot  of  air  at  14  Ibs.  and 
60°  is  .07277,  and  .07277  X  2  =  .14554.  The  absolute  t  is 
460  +  60  =  520,  and  r  =  6.0. 

Then  in  Table  I,  column  11,  opposite  r  —  6  we  find  95.271, 
whence 

W  =  95.271  X  520  X  .14554  =  7208. 

The  difference  in  the  two  results  is  due  to  dropping  off  the 
fraction  in  temperature. 

Art.  2.  Temperature  Varying. 

The  conditions  are  said  to  be  adiabatic  when,  during  com- 
pression or  expansion,  no  heat  is  allowed  to  enter  in,  or 
escape  from,  the  air  although  the  temperature  in  the  body 
of  confined  air  changes  radically  during  the  process. 

Physicists  have  proved  that  under  adiabatic  conditions 
the  following  relations  hold: 


and  since  for  one  pound  of  air  at  32°  F.  pv  =  26,214  and  t  = 
492.6,  we  get  for  one  pound  at  any  pressure,  volume  and 
temperature, 

pv  =  53.17  1.  (4) 

While  formulas  (3)  and  (4)  are  very  important,  they  do  not 
apply  to  the  actual  conditions  under  which  compressed 
air  is  worked,  for  in  practice  we  get  neither  isothermal  nor 
adiabatic  conditions  but  something  intermediate. 

For  such  conditions  physicists  have  discovered  that  the 
following  holds  nearly  true: 

=  pxvxn  =  p2vf,  (5) 


4  COMPRESSED  AIR 

sub  x  indicating  any  intermediate  stage  and  the  exponent  n 
varying  between  1  and  1.41  according  to  the  effectiveness 
of  the  cooling  in  case  of  compression  or  the  heating  in  case  of 
expansion.  From  this  basic  formula  (5)  the  formulas  for 
work  must  be  derived. 

As  in  Art.  (1)  dW  =  pxdvx  =  V^^~  =  Pi»in  <Xr 
Therefore 


Now  since  p&i"  X  v2l~n  =  p2V2nX  v2l~n  =  p2v2  and 
=  pn>i  the  expression  becomes 

—  P\v\ 


n—  I 

which  represents  the  work  done  in  compression  or  expansion 
between  B  and  C,  Fig.  1.  To  this  must  be  added  the  work 
of  expulsion,  p2v2,  and  from  it  must  be  subtracted  the  work 
done  by  the  air  entering  behind  the  piston,  piVi.  Hence  the 
whole  net  work  done  in  one  stroke  is 

(6) 


n  — 

=  —^—r  (p&2  -  pi»i).  (7) 

n  —  1 

Equation  (7)  is  in  convenient  working  form  and  may  be  used 
when  the  data  are  in  pressures  and  volumes,  but  it  is  common 
to  express  the  compression  or  expansion  in  terms  of  r.  For 
such  cases  a  convenient  working  formula  is  gotten  as  follows: 

From  Eq.  (5) 


A  i  T>2        Vin       j_i         f  V\ 

Also  r  =  ^  =  —  ,    therefore    —  =  rn , 

PI      v2  v2 

n-l          n~1  n-l 

and  — —  =  T  n  >    therefore     '"-"-  —  '"--»-*•  n 


and  Eq.  (7)  becomes  W  =  —3—  ftAir  *     -  ij.  (8) 


FORMULAS   FOR  WORK  5 

The  most  common  uses  of  equations  (7)  and  (8)  are  when 
air  is  compressed  from  free  air  conditions,  then  pi  and  v\  be- 
come pa  and  va.  This  case  must  be  carefully  distinguished 
from  the  case  of  incomplete  expansion  as  presented  in  Art.  3. 

In  perfectly  adiabatic  conditions  n  =^1.41,  but  in  practice 
the  compressor  cylinders  are  water-jacketed  and  thereby 
part  of  the  heat  of  compression  is  conducted  away,  so  that 
n  is  less  than  1.41.  For  such  eases  Church  assumes  n  =  1.33 
and  Unwin  assumes  n  =  1.25.  Undoubtedly  the  value 
varies  with  size  and  proportions  of  cylinders,  details  of 
water-jacketing,  temperature  of  cooling  water  and  speed  of 
compressors.  Hence  precision  in  the  value  of  n  is  not  prac- 
ticable. Fortunately  the  work  does  not  vary  as  much  as 
n  does. 

For  one  pound  of  air  at  initial  temperature  of  60°  F. 
Eq.  (8)  gives  in  foot-pounds, 

When  n  =  1.41,  W  =  95,193  (r°-29  -  1).  (8b) 

When  n  =  1.25,  W  =  138,405  (r°-2  -  1).  (8c) 

Common  log  of    95,193  =  4.978606. 
Common  log  of  138,405  =  5.141141. 

The  above  special  values  will  be  found  convenient  for 
approximate  computations.  For  compound  compression 
see  Art.  12. 

If  in  Eq.  (8)  we  substitute  for  pv  its  value,  53.17  t,  for 
one  pound,  we  get 

f/         \  /  *-i        Yl 

W=    (-     -)53.17\rn    -l]  \Xt  =  Kt,          (8d) 
Lv*     i/  J 

/   n-l  \ 

where  k  =  -  n—  X  53.17^  n     -  l)  - 

Table  I  gives  values  of  K  for  n  =  1.25  and  n  =  1.41  and 
for  values  of  r  up  to  10,  varying  by  one-tenth.  The  theoretic 
work  in  any  case  is  K  X  Q  X  t,  where  Q  is  the  number  of 
pounds  passed  and  t  is  the  absolute  lower  temperature. 
Further  explanation  accompanies  the  table. 

The  difference  between  isothermal  and  adiabatic  compres- 
sion (and  expansion)  can  be  very  clearly  shown  graphically 


6 


COMPRESSED  AIR 


as  in  Fig.  2.  In  this  illustration  the  terminal  points  are 
correctly  placed  for  a  ratio  of  5  for  both  the  compression  and 
expansion  curve. 


Fig.  2. 


Note'that  in  the  compression  diagram  (a),  the  area  between 
the  two  curves  aef  represents  the  work  lost  in  compres- 
sion due  to  heating,  and  the  area  between  the  two  curves 
aeghb  in  (6)  represents  the  work  lost  by  cooling  during 
expansion.  The  isothermal  curve,  a  e,  will  be  the  same  in 
the  two  cases. 

Such  illustrations  can  be  readily  adapted  to  show  the 
effect  of  reheating  before  expansion,  cooling  before  compres- 
sion, heating  during  expansion,  etc. 

Example  2a.  What  horse  power  will  be  required  to  com- 
press 1000  cubic  feet  of  free  air  per  minute  from  pa  =  14.5 
to  a  gage  pressure  =  80,  when  n  =  1.25  and  initial  tempera- 
ture =  50°  F.? 

Solution.  From  Table  II,  interpolating  between  40°  and 
60°  the  weight  of  one  cubic  foot  is  .07686  and  the  weight  of 
1000  is  76.86  -.  The  r  from  above  data  is  6.5.  Then  in 


FORMULAS  FOR  WORK  7 

Table  I  opposite  r  =  6.5  in  column  9  we  find  .3658.     Then 


Horse  power  =  .3658  X 


100 


X  510  =  143. 


The  student  should  check  this  result  bj£,Eq.  (8)  or  (8d)  with- 
out the  aid  of  the  table. 

Art.  3.  Incomplete  Expansion. 

When  compressed  air  is  applied  in  an  engine  as  a  motive 
power  its  economical  use  requires  that  it  be  used  expansively 
in  a  manner  similar  to  the  use  of  steam.  But  it  is  never 
practicable  to  expand  the  air  down  to  the  free  air  pressure, 
for  two  reasons  :  —  first,  the  increase  of  volume  in  the  cylin- 
ders would  increase  both  cost  and  friction  more  than  could 
be  balanced  by  the  increase  in  power;  and  second,  unless 
some  means  of  reheating  be  provided,  a  high  ratio  of  expan- 
sion of  compressed  air  will  cause  a  freezing  of  the  moisture 
in  and  about  the  ports. 

The  ideal  indicator  diagram  for  incomplete  expansion  is 
shown  in  Fig.  3.  In  such  diagrams  it  is  convenient  and 


Fig.  3. 

simplifies  the  demonstrations  to  let  the  horizontal  length 
represent  volumes.  In  any  cylinder  the  volumes  are  pro- 
portional to  the  length. 

Air  at  pressure  p2  is  admitted  through  that  part  of  the 
stroke  represented  by  v2  —  thence  the  air  expands  through 
the  remainder  of  the  stroke  represented  by  vi,  the  pressure 
dropping  to  pi.  At  this  point  the  exhaust  port  opens  and 
the  pressure  drops  to  that  of  the  free  air.  The  dotted  por- 
tion would  be  added  to  the  diagram  if  the  expansion  should 
be  carried  down  to  free  air  pressure. 


8  COMPRESSED  AIR 

To  write  a  formula  for  the  work  done  by  the  air  in  such  a 
case  we  will  refer  to  Eq.  (6)  and  its  derivation.  In  the  case 
of  simple  compression  or  complete  expansion  it  is  correctly 
written 


which  would  give  work  in  the  case  represented  by  Fig.  1 
when  there  is  a  change  of  temperature,  but  in  such  a  case  as 
is  represented  by  Fig.  3  the  equation  must  be  modified  thus  : 

pM,  (9) 


n  —  1 

the  reason  being  apparent  on  inspection. 

In  numerical  problems  under  Eq.  (9)  there  will  be  known 
p2v2,  n,  and  either  pi  or  VL     The  unknown  must  be  computed 

from  the  relations  from  Eq.  (5)  : 

i 


fv2\n  ivzY 

Pi  =  Pz  ( -  )      or    vi  =  v2  (  — )  • 

W  W 


Example  3a.  A  compressed-air  motor  takes  air  at  a  gage 
pressure  =  100  Ibs.  and  works  with  a  cut-off  at  J  stroke. 
What  work  (ft.-lbs.)  will  be  gotten  per  cu.  ft.  of  compressed 
air,  assuming  free  air  pressure  =  14.5  Ibs.  and  n  =  1.41  ? 

Solution.  Applying  Eq.  (9)  and  noting  that  all  pressures 
are  to  be  multiplied  by  144  and  that  the  pressure  at  end  of 


/JA1.41 

stroke  =  pi  =  114.5  f1)      =  16.3  and  that^i  =  4z;2,  we  get 

=  25,444. 


Art.  4.  Effect  of  Clearance  :  In  Compression. 

It  is  not  practicable  to  discharge  all  of  the  air  that  is 
trapped  in  the  cylinder;  there  are  some  pockets  about  the 
valves  that  the  piston  cannot  enter,  and  the  piston  must  not 
be  allowed  to  strike  the  head  of  the  cylinder.  This  clearance 
can  usually  be  determined  by  measuring  the  water  that  can 
be  let  into  the  cylinder  in  front  of  the  piston  when  at  the  end 
of  its  stroke  ;  but  the  construction  of  each  compressor  must 


FORMULAS  FOR  WORK  9 

be  studied  before  this  can  be  undertaken  intelligently,  and 
it  is  not  done  with  equal  ease  in  all  machines. 

To  formulate  the  effect  of  this  clearance  in  the  operation 
of  the  machine, 

Let  v  =  volume  of  piston  displacement  (  =  area  of  piston 
X  length  of  stroke)  , 

Let  cv  =  clearance,  c  being  a  percentage. 

Then  v  +  c  v  is  the  volume  compressed  each  stroke.  But 
the  clearance  volume  cv  will  expand  to  a  volume  rev  as  the 
piston  recedes,  so  that  the  fresh  air  taken  in  at  each  stroke 
will  be  v  +  cv  —  rev,  and  the  volumetric  efficiency  will  be 

v  =  \  +  c(\-r).  (10) 


When  Ev  =  0  c  =  --  and  no  air  will  be  discharged. 

Theoretically  (as  the  word  is  usually  used)  clearance  does 
not  cause  a  loss  of  work,  but  practically  it  does,  insomuch  as 
it  requires  a  larger  machine,  with  its  greater  friction,  to  do 
a  given  amount  of  effective  work. 

Example  4a.  A  compressor  cylinder  is  12"  diam.  X  16" 
stroke.  The  clearance  is  found  to  hold  1^  pints  of  water 

=  -7—  X  231  =36    cubic   inches;    therefore   c  = 


113  X  16 
=  0.02.  /f 

Then  by  Eq.  (10)  when  r  =  7 

E  =  1  +  0.02  (1  -  7)  =  88%. 

Such  a  condition  is  not  abnormal  in  small  compressors,  and 
the  volumetric  efficiency  is  further  reduced  by  the  heating 
of  air  during  admission  as  considered  in  Art.  6. 

Art.  5.  Effect  of  Clearance  and  Compression  in  Expansion 
Engines. 

Fig.  4  is  an  ideal  indicator  diagram  illustrating  the  effect 
of  clearance  and  compression  in  an  expansion  engine. 

In  this  diagram  the  area  E  shows  the  effective  work,  D 
the  effect  of  clearance,  B  the  effect  of  back  pressure  of  the 
atmosphere  and  C  the  effect  of  compression  on  the  return 
stroke. 


10 


COMPRESSED  AIR 


The  study  of  effect  of  clearance  in  an  expansion  engine 
differs  from  the  study  of  that  in  compression,  due  to  the 
fact  that  the  volume  in  the  clearance  space  is  exhausted 
into  the  atmosphere  at  the  end  of  each  stroke. 


Fig.  4. 

If  the  engine  takes  full  pressure  throughout  the  stroke  the 
air  (or  steam)  in  the  clearance  is  entirely  wasted;  but  when 
the  air  is  allowed  to  expand  as  illustrated  in  the  diagram  some 
useful  work  is  gotten  out  of  the  air  in  the  clearance  during 
the  expansion. 

The  loss  due  to  clearance  in  such  engine  is  modified  by  the 
amount  of  compression  allowed  in  the  back  stroke.  If  the 
compression  pc  =  p2,  the  loss  of  work  due  to  clearance  will  be 
nothing,  but  the  effective  work  of  the  engine  will  be  consid- 
erably reduced,  as  will  be  apparent  by  a  study  of  a  diagram 
modified  to  conform  to  the  assumption. 

While  the  formula  for  work  that  includes  the  effect  of 
clearance  and  compression  will  not  be  often  used  in  practice 
its  derivation  is  instructive  and  gives  a  clear  insight  into 
these  effects. 

The  symbols  are  placed  on  the  diagram  and  will  not  need 
further  definition. 

The  effective  work  E  will  be  gotten  by  subtracting  from 
the  whole  area  the  separate  areas  B,  C  and  D.  From  Art.  2, 
after  making  the  proper  substitutions  for  the  volumes,  there 
results 

'pz(c-{-  k)—  pi  (1  +  c) 
n-l 


Total  area 


=  1  1 


+P,(c  +  *) 


]. 


FORMULAS  FOR  WORK  11 


Area  B  =  lpaj 
Area  D  =  Ip2c, 

Area  C  =  iP'*  ~  P 


n~i 

Subtracting  the  last  three  from  the  first  and  reducing  their 
results: 

Work        1 


Al       n—l 

=  Mean  effective  pressure. 

The  actual  volume  ratio  before  and  after  expansion  is 

Vi          CVi  +  Vi          C  +  1 

This  is  the  ratio  with  which  to  enter  Table  I  to  get  r  and  t 
and  from  r  the  unknown  pressure  p\.     Similarly  for  the 

s* 

compression  curve  the  ratio  of  volumes  is  -,  and  pc  can  be 

o 

found  as  indicated  above. 

Art.  6.   Effect  of  Heating  Air  as  it  Enters  Cylinders. 

When  a  compressor  is  in  operation  all  the  metal  exposed 
to  the  compressed  air  becomes  hot  even  though  the  water 
jacketing  is  of  the  best.  The  entering  air  comes  in  contact 
with  the  admission  valves,  cylinder  head  and  walls  and  the 
piston  head  and  piston  rod,  and  is  thereby  heated  to  a  very 
considerable  degree.  In  being  so  heated  the  volume  is  in- 
creased in  direct  proportion  to  the  absolute  temperature 
(see  Eq.  (5) ),  since  the  pressure  may  be  assumed  constant 
and  equal  that  of  the  atmosphere.  Hence  a  volume  of 
cool  free  air  less  than  the  cylinder  volume  will  fill  it  when 
heated.  This  condition  is  expressed  by  the  ratio 

%*  =  f     or    va  =  vcf, 
vc       tc  tc 

where  vc  and  tc  represent  the  cylinder  volume  and  tempera- 
ture.    The  volumetric  efficiency  as  effected  by  the  heating  is 


12  COMPRESSED  AIR 

Example  6a.  Suppose  in  Example  4a  the  outside  free  air 
temperature  is  60°  F.  and  in  entering  the  temperature  rises 
to  160°  F.,  then 

460  +  60 


tc      460  +  160 


Then  the  final  volumetric  efficiency  would  be  88  X  84  = 
74%  nearly. 

The  volumetric  efficiency  of  a  compressor  may  be  further 
reduced  by  leaky  valves  and  piston. 

In  Arts.  4  and  6  it  is  made  evident  that  the  volumetric 
efficiency  of  an  air  compressor  is  a  matter  that  cannot  be 
neglected  in  any  case  where  an  installation  is  to  be  intelli- 
gently proportioned.  It  should  be  noted  that  the  volu- 
metric efficiency  varies  with  the  various  makes  and  sizes 
of  compressors  and  that  the  catalog  volume  rating  is  always 
based  on  the  piston  displacement. 

These  facts  lead  to  the  conclusion  that  much  of  the  uncer- 
tainty of  computations  in  compressed-air  problems  and  the 
conflicting  data  recorded  is  due  to  the  failure  to  determine 
the  actual  amount  of  air  involved  either  in  terms  of  net 
volume  and  temperature  or  in  pounds. 

Methods  of  determining  volumetric  efficiency  of  air  com- 
pressors are  given  in  Chapter  III. 

The  loss  of  work  due  to  the  air  heating  as  it  enters  the 
compressor  cylinder  is  in  direct  proportion  to  the  loss  of 
volumetric  efficiency  due  to  this  cause.  In  Example  6a 
only  84%  of  the  work  done  on  the  air  is  effective. 

By  the  same  law  any  cooling  of  the  air  before  entering  the 
compressor  effects  a  saving  of  power.  See  Art!  9. 

Art.  7.  Change  of  Temperature  in  Compression  or  Ex- 
pansion. 

Eq.  (4)  may  be  written 

Pivi  =  cti;  p2v2  =  ct2 

and  Eq.  (5)  may  be  factored  thus, 


Substituting  we  get 

ctlVln~l  =  ct2v2n~ 


FORMULAS  FOR  WORK  13 

Whence  fe^feY"  (11) 


and  Ir-'tilrl    ..  =  ^  n  9  (lla) 

since  from  Eq.  (5) 

'Pi 

It  is  possible  to  compu  ten  from  Eq.  (11)  by  controlling  the 
Vi  and  t>2  and  measured  ti  and  22- 

Table  I,  columns  5  and  6,  is  made  up  from  Eq.  (lla)  and 
columns  3  and  4  from  Eq.  (5)  as  just  written. 

Example  7.  What  would  be  the  temperature  of  air  at  the 
end  of  stroke  whenr  =  7  and  initial  temperature  =  70°  F.? 

Solution.    Referring  to  Table  I  in  line  with  r=  7  note  that 

1.4758  when  n=  1.25 

/.   t2  =  (460  +  70)  X  1.4758  -  460  =  322°  F. 
1.7585  when  n  =  1.41 

/.   t2  =  (460  +  70)  X  1.7585  -  460  =  472°  F. 

From  the  same  table  the  volume  of  one  cubic  foot  of  free 
air  when  compressed  and  still  hot  would  be  respectively  0.21 
and  0.25,  while  when  the  compressed  air  is  cooled  back  to 
70°  its  volume  would  be  0.143. 

Art.  8.  Density  at  Given  Temperature  and  Pressure. 

By  Eq.  (4)  pv  =  53.17  t  for  one  pound,  and  the  weight  of 
one  cubic  foot 

=  w  =  l= — 2 —  (12) 

v       53.17* 

Note  that  p  must  be  the  absolute  pressure  in  pounds  per 
square  foot,  and  t  the  absolute  temperature.  When  gage 
pressures  are  used  and  ordinary  Fahrenheit  temperature 
the  formula  becomes 

144  ,,..,., 

F> 


Table  III  is  made  up  from  Eq.  (12). 


14  COMPRESSED  AIR 

Art.  9.    Cooling  Water  Required. 

In  isothermal  changes,  since  pv  is  constant,  evidently 
there  is  no  change  in  the  mechanical  energy  in  the  body  of 
air  as  measured  by  the  absolute  pressure  and  using  the  term 
"mechanical  energy  "to  distinguish  from  heat  energy.  Hence 
evidently  all  the  work  delivered  to  the  air  from  outside  must 
be  abstracted  from  the  air  in  some  other  form,  and  we  find 
it  in  the  heat  absorbed  by  the  cooling  water.  Therefore, 

(B.T.U's) 

of  work  done  on  compressed  air  =  35.5  log  r  (B.T.U's)  per 
pound  of  air  compressed  from  temperature  of  60°  F.  If  the 
water  is  to  have  a  rise  of  temperature  T°  (T  being  small,  else 
the  assumption  of  isothermal  changes  will  not  hold),  then 

pv  oge  r  _  poun(js  of  water  required  in  same  time. 

Example  8a.  How  many  cubic  feet  of  water  per  minute  will 
be  required  to  cool  1000  cubic  feet  of  free  air  per  minute, 
air  compressed  from  pa  =  14.2  to  pg  =  90°  gage,  initial  tem- 
perature of  air  =  50°  F.  and  rise  in  temperature  of  cooling 
water  =  25°  ? 

Solution: 


144  X  14.2  X  1000  X 

780X25X62.5  -  24  cu.  ft.  per  min. 

It  is  practically  possible  to  attain  nearly  isothermal  con- 
ditions by  spraying  cool  water  into  the  cylinder  during 
compression.  In  such  a  case  this  article  would  enable  the 
designer  to  compute  the.  quantity  of  water  necessary  and 
therefrom  the  sizes  of  pipes,  pumps,  valves,  etc. 

Art.  10.   Reheating  and  Cooling. 

In  any  two  cases  of  change  of  state  of  a  given  weight  of 
air,  assuming  the  ratio  of  change  in  pressure  to  be  the  same, 
the  work  done  (in  compression  or  expansion)  will  be  directly 
proportional  to  the  volume,  as  will  be  evident  by  examina- 
tion of  the  formulas  for  work.  Also  at  any  given  pressure 
the  volumes  will  be  directly  proportional  to  the  absolute  tem- 
peratures. Hence  the  work  done  either  in  compression  or 


FORMULAS  FOR  WORK  15 

expansion  (ratio  of  change  in  pressures  being  the  same  in  each 
case)  will  be  directly  proportional  to  the  absolute  initial  tem- 
peratures. 

Thus  if  the  temperature  of  the  air  in  the  intake  end  of  one 
compressor  is  160°  F.  and,  in  another  50°  F.,  the  work  done 
on  equal  weights  of  air  in  the  two  cases  will  be  in  the  pro- 
portion of  460  +  150  to  460  +  50,  or  1.2  to  1;  that  is,  the 
work  in  the  first  case  is  20%  more  than  that  in  the  second 
case.  This  is  equally  true,  of  course,  for  expansion. 

The  facts  above  stated  reveal  a  possible  and  quite  practi- 
cable means  of  great  economy  of  power  in  compressing  air 
and  in  using  compressed  air. 

The  opportunities  for  economy  by  cooling  for  compression 
are  not  as  good  as  in  heating  before  the  application  in  a 
motor,  but  even  in  compression  marked  economy  can  be 
gotten  at  almost  no  cost  by  admitting  air  to  the  compressor 
from  the  coolest  convenient  source,  and  by  the  most  thorough 
water-jacketing  with  the  coolest  water  that  can  be  conven- 
iently obtained. 

In  all  properly  designed  compressor  installations  the  air  is 
supplied  to  the  machine  through  a  pipe  from  outside  the 
building  to  avoid  the  warm  air  of  the  engine  room.  In 
winter  the  difference  in  temperature  may  exceed  100°,  and 
this  simple  device  would  reduce  the  work  of  compression  by 
about  20%.  For  the  effect  of  intercoolers  and  interheaters 
see  Art.  .12  on  compounding. 

By  reheating  before  admitting  air  to  a  compressed-air 
engine  of  any  kind  the  possibilities  of  effecting  economy 
of  power  are  greater  than  in  cooling  for  compression,  the 
reason  being  that  heating  devices  are  simpler  and  less  costly 
than  any  means  of  cooling  other  than  those  cited  above. 

The  compressed  air  passing  to  an  engine  can  be  heated  to 
any  desired  temperature;  the  only  limit  is  that  temperature 
that  will  destroy  the  lubrication.  Suppose  the  normal 
temperature  of  the  air  in  the  pipe  system  is  60°  F.  and  that 
this  is  heated  to  300°  F.  before  entering  the  air  engine,  then 
the  power  is  increased  46%.  Reheating  has  the  further 
advantage  that  it  makes  possible  a  greater  ratio  of  expansion 
without  the  temperature  reaching  freezing  point. 


16  COMPRESSED  AIR 

The  devices  for  reheating  are  usually  a  coil  or  cluster  of 
pipes  through  which  the  air  passes  while  the  pipe  is  exposed 
to  the  heat  of  combustion  from  outside.  Ordinary  steam 
boilers  may  be  used,  the  air  taking  the  place  of  the  steam  and 
water. 

Unwin  suggests  reheating  the  air  by  burning  the  fuel  in 
the  compressed  air  as  suggested  in  the  cut. 


°f-^=F- 

Combustion 

jut  *LJP  ^  j,  a  *  !*_«  ,«, 

F^ 

Hot  Air 


Gas,  Liquid  or 
Powdered  Fuel 

Cold  Air, 

Li . 

Fig.  4  A. 

Even  when  the  details  are  worked  out  such  a  device  would 
be  simple  and  inexpensive.  The  theoretic  advantages  of 
such  a  device  are  that  all  the  heat  would  go  into  the  air, 
the  gases  of  combustion  (if  solid  or  liquid  fuel  be  used)  would 
increase  the  volume,  and  the  combustion  occurring  in  com- 
pressed air  would  be  very  complete. 

The  author  has  no  knowledge  of  any  such  devices  having 
been  used  in  practice. 

The  power  efficiency  of  the  fuel  used  in  reheaters  is  very 
much  greater  than  that  of  the  fuel  used  in  steam  boilers. 
Unwin  states  that  it  is  five  or  six  times  as  much.  The  chief 
reason  is  that  none  of  the  heat  is  absorbed  in  evaporation 
as  in  a  steam  boiler. 

In  many  of  the  applications  of  compressed  air  reheating  is 
impracticable,  and  efficiency  is  secondary  to  convenience  — 
but  in  large  fixed  installations,  such  as  mine  pumps,  reheat- 
ing should  be  applied. 

Art.  11.   Compounding. 

In  steam-engine  designs  compounding  is  resorted  to  to 
economize  power  by  saving  steam,  while  in  air  compressors 
and  compressed-air  engines  compounding  is  resorted  to  for 
the  twofold  purpose  of  economizing  power  and  controlling 
temperature,  both  objects  being  accomplished  by  reducing 
the  extreme  change  of  temperature.  The  economic  prin- 


FORMULAS  FOR  WORK 


17 


ciples  involved  in  compound  steam  engines  and  in  com- 
pound air  engines  are  quite  different,  the  reasons  underlying 
the  latter  being  much  more  definite. 

The  air  is  first  compressed  to  a  moderate  ratio  in  the 
low-pressure  cylinder,  whence  it  is  discharged  into  the  "  inter- 
cooler,"  where  most  of  the  heat  developed  in  the  first  stage 
is  absorbed  and  thereby  the  volume  materially  reduced,  so 
that  in  the  second  stage  there  will  be  less  volume  to  com- 
press and  a  less  injurious  temperature. 

The  changes  occurring  and  the  manner  in  which  economy 
is  effected  in  compression  may  be  most  easily  understood 
by  reference  to  Fig.  5,  which  represents  ideal  indicator 
diagrams  from  the  two  cylinders,  superimposed  one  over 
the  other,  the  scale  being  the  same  in  each,  the  dividing 
line  being  kb. 


e     d  f    g 


Fig.  5. 

In  this  diagram,  Fig.  5, 

dbk  is  the  compression  line  in  the  first-stage  or  low-pressure 
cylinder, 

cds  is  the  compression  line  in  the  second-stage  or  high-pres- 
sure cylinder, 

be  is  the  reduction  of  volume  in  the  intercooler, 


18 


COMPRESSED  AIR 


abf  would  be  the  pressure  line  if  no  intercooling  occurred, 
The  area  cdfb  is  the  work  saved  by  the  intercooler, 
ace  would  be  the  compression  line  for  isothermal  compres- 
sion, 

aug  would  be  the  compression  line  for  adiabatic  compres- 
sion. 

The  diagram  Fig.  5  is  correctly  proportioned  for  r  =  6. 

Fig.  6  is  a  diagram  drawn  in  a  manner  similar  to  that  used 
in  Fig.  5  and  is  to  illustrate  the  changes  and  economy  effected 
by  compounding  with  heating  when  compressed  air  is  applied 
in  an  engine.  It  is  assumed  that  the  air  is  " preheated," 
that  is,  heated  once  before  entering  the  high-pressure  cylinder 
and  again  heated  between  the  two  cylinders. 


Fig.  6. 

In  this  diagram,  Fig.  6, 

se  is  the  volume  of  compressed  air  at  normal  temperature, 
sf  is  the  volume  of  compressed  air  after  preheating, 
fc  is  the  expansion  line  in  the  high-pressure  cylinder, 
cb  is  the  increase  of  volume  in  the  interheater, 
by  is  the  expansion  line  in  low-pressure  cylinder, 
ezq  would  be    the  adiabatic    expansion   line    without    any 

heating, 

efcz  is  work  gained  by  preheating, 
cbyx  is  work  gained  by  interheating. 


FORMULAS  FOR  WORK  19 

In  no  case  is  it  economical  to  expand  down  to  atmospheric 
pressure.  Hence  the  diagram  is  shown  cut  off  with  pressure 
still  above  that  of  free  air. 

The  diagram  Fig.  6  is  proportioned  for  preheating  and  re- 
heating 300°  F. 

Art.  12.     Proportions  Jfor  Compounding. 

It  is  desirable  that  equal  work  be  done  in  each  stage  of 
compounding.  If  this  condition 'be  imposed,  Eq.  (8)  indi- 
cates that  the  r  must  be  the  same  in  each  stage,  for  on  the 
assumption  of  complete  intercooling  the  product  pv  will 
be  the  same  at  the  beginning  of  each  stage. 

If  then  7*1  be  the  ratio  of  compression  in  the  first  stage, 
the  pressure  at  end  of  first  stage  will  be  ripa  =  pi,  and  the 
pressure  at  end  of  second  stage  =  r±pi  =  r-?pa  —  p2,  and 
similarly  at  end  of  third  stage  the  pressure  will  be  ps  =  r^  pa, 
or 

In  two-stage  work     n  =  (2i  )2  =  r2* 
In  three-stage  work  7*1 

Let  Vi  =  free  air  intake  per  stroke  in  low-pressure  cylinder 

or  first  stage, 

v2  =  piston  displacement  in  second  stage, 
v3  =  piston  displacement  in  third  stage, 
TI  =  ratio  of  compression  in  each  cylinder. 
Then,  assuming  complete  intercooling, 

V\  j  ?>2  Vl 

v%  =  —     ana    Vz  =  —  =  —, 
or  and    -  =  — • 

The  length  of  stroke  will  be  the  same  in  each  cylinder: 
therefore  the  volumes  are  in  the  ratio  of  the  squares  of 
diameters,  or 


Hence  d2  =        and    d,  =      -  (12b) 


20  COMPRESSED  AIR 

If  the  intention  to  make  the  work  equal  in  the  different 
cylinders  be  strictly  carried  out  it  will  be  necessary  to  make 
the  first-stage  cylinder  enough  larger  to  counteract  the 
effect  of  volumetric  efficiency.  Thus  if  volumetric  efficiency 
be  75%,  the  volume  (or  area)  of  the  intake  cylinder  should 
be  one-third  larger.  Note  that  the  volumetric  efficiency  is 
chargeable  entirely  to  the  intake  or  low-pressure  cylinder. 
Once  the  air  is  caught  in  that  cylinder  it  must  go  on. 

Example  12.  Proportion  the  cylinders  of  a  compound  two- 
stage  compressor  to  deliver  300  cu.  ft.  of  free  air  per  minute 
at  a  gage  pressure  =  150.  Free  air  pressure  =  14.0, 
R.P.M.  =  100,  stroke  18",  piston  rod  If"  diameter,  volu- 
metric efficiency  =  75%. 

Solution.  From  the  above  data  the  net  intake  must  be 
3  cu.  ft.  per  revolution.  Add  to  this  the  volume  of  one  piston 
rod  stroke  (  =  .025  cu.  ft.)  and  divide  by  2.  This  gives 
the  volume  of  one  piston  stroke  1.512.  The  volume  of  one 

foot  of  the  cylinder  will  be  —  X  1.512  =  1  .008  cu.  ft.     From 

18 

Table  X  the  nearest  cylinder  is  14"  diam.,  the  total  ratio  of 
compression  =-  -=  11.71,  and  the  ratio  in  each  stage 

is  (11.71)*=  3.7  =  n,  and  by  (12b) 

d2  =  —  ~  =  —  —  =  7.  3",  say  7f",  for  the  high-pressure  cylinder. 
(ri)a  1-92 

Now  we  must  increase  the  low-pressure  cylinder  by  one- 
third  to  allow  for  volumetric  efficiency.  The  volume  per 
foot  will  then  be  1.344,  which  will  require  a  cylinder  about 
15f  "  diameter.  Note  that  the  diameter  of  the  high-pressure 
cylinder  will  not  be  affected  by  the  volumetric  efficiency. 

Art.  13.     Work  in  Compound  Compression. 

Assuming  that  the  work  is  the  same  in  each  stage,  Eq.  (8) 
can  be  adapted  to  the  case  of  multistage  compression  thus  :  — 

In  two-stage  work 


/     n-l  \ 

=~         -paVaVl   n     -I]    X2 

-  l)  X  2.  (13a) 


W=~         -paVaVl   n     -I]    X2  (13) 

I 


n  —  1 


FORMULAS  FOR  WORK  21 

In  three-stage  work 

-  l      X  3  (13b) 

(13C) 


=  -^PaVaVt^-  l)  X  3. 
Tl  —  1 


Note  that  r2  =  —  and  r3  =  ^  and  also  that  pava 

Pa  Pa 

,  etc.,  assuming  complete  intercooling. 
Laborious  precision  in  computing  the  work  done  on  or  by 
compressed  air  is  useless,  for  there  are  many  uncertain  and 
changing  factors:  n  is  always  uncertain  and  changes  with 
the  amount  and  temperature  of  the  jacket  water,  the  volu- 
metric efficiency,  or  actual  amount  of  air  compressed,  is 
usually  unknown,  the  value  of  pa  varies  with  the  altitude, 
and  r  is  dependent  on  pa. 

Art.  14.    Work  under  Variable  Intake  Pressure. 

There  are  some  cases  where  air  compressors  operate  on  air 
in  which  the  intake  pressure  varies  and  the  delivery  pressure 
is  constant.  This  is  true  in  case  of  exhaust  pumps  taking  air 
out  of  some  closed  vessels  and  delivering  it  into  the  atmos- 
phere. It  is  also  the  condition  in  the  "  return-air"  pumping 
system  in  which  one  charge  of  air  is  alternately  forced  into 
a  tank  to  drive  the  water  out  and  then  exhausted  from  the 
tank  to  admit  water.  For  full  mathematical  discussion  of 
this  pump  see  Trans.  Am.  So.  C.  E.,  Vol.  54,  p.  19.  The 
following  formulas  and  others  more  complex  were  first 
worked  out  to  apply  to  that  pumping  system. 

In  such  cases  it  is  necessary  to  determine  the  maximum 
rate  of  work  in  order  to  design  the  motive  power. 

First  assume  the  operation  as  being  isothermal.  Then 
in  Eq.  (1),  viz. 

W  =  pxV  loge  2t 
Px 

px  is  variable,  while  v  and  pi  are  constant.  In  this  formula 
W  becomes  zero  when  px  is  zero  and  again  when  px  =  pi, 
since  log  1  is  zero.  To  find  when  the  work  is  maximum, 
differentiate  and  equate  to  zero;  thus  differential  of 


22  COMPRESSED   AIR 

v  (px  log  pi  -  px  log  px)=v\  log  pidpx  -  (px-&L  +  log  pxdpx\ 
Equate  this  to  zero  and  get  log  pi  =  1  +  log  Px, 
or  log  2i  =  i,    therefore  ^  =  e  =  2.72. 

Px  Px 

That  is,  when  r  =  2.72  the  work  is  a  maximum. 

When  the  temperature  exponent  n  is  to  be  considered  the 
study  must  be  made  in  Eq.  (8),  viz. 


Differentiating  this  with  respect  to  px  and  equating  to  zero, 

n-i 

the  condition  for  maximum  work  becomes!  ^  )       —  n.     Insert 

W 
this  in  (8)  and  the  reduced  formula  becomes 


From  the  above  expressions  for  maximum    the   following 
results  : 

When  n  =  1.41  the  maximum  occurs  when  r  —  3.26. 

When  n  =  1.25  the  maximum  occurs  when  r  —  3.05. 

When  n  =  1.      the  maximum  occurs  when  r  =  2.72. 
In  practice  r  =  3  will  be  a  safe  and  convenient  rule. 

Exercise  14a.  Air  is  being  exhausted  out  of  a  tank  by  an  ex- 
haust pump  with  capacity  =  1  cu.  ft.  per  stroke.  At  the  be- 
ginning the  pressure  in  the  tank  is  that  of  the  atmosphere  = 
14.7  Ibs.  per  sq.  in.  Assume  the  pressure  to  drop  by  intervals 
of  one  pound  and  plot  the  curve  of  work  with  px  as  the 
horizontal  ordinate  and  W  as  the  vertical,  using  the  formula 

W  =  pxvlog^- 

Px 

Exercise  146.  As  in  14a  plot  the  curve  by  Eq.  (8)  with 
n  =  1.25. 

Art.  15.    Exhaust  Pumps. 

In  designing  exhaust  pumps  the  following  problems  may 
arise. 


FORMULAS  FOR  WORK  23 

Given  a  closed  tank  and  pipe  system  of  volume  V  under 
pressure  p0  and  an  exhaust  pump  of  stroke  volume  v,  how 
many  strokes  will  be  necessary  to  bring  the  pressure  down 
topm? 

The  analytic  solution  is  as  fqllows,  assuming  isothermal 
conditions  in  the  volume  V. 

The  initial  product  of  pressure  by  volume  is  p0V.  After 
the  first  stroke  of  the  exhaust  pump  this  air  has  expanded 
into  the  cylinder  of  the  pump  and  pressure  has  dropped  to 
Pi  under  the  law  that  pressure  by  volume  is  constant. 

Hence  (V  +  v)  p-L  =p0V,  or  pi  =  -^  —  at  end  of  first  stroke, 


at  end  of  second  stroke, 

<F  +  .)p.-*7,      or  P3  =  P2^-.  =  PO(^-J 
at  end  of  third  stroke,  etc. 

/   v  \m  log  t9 

Finally       pm  =  p0  (  — —  -  and    m  = — ft-  .         (14) 


m  is  the  required  number  of  strokes. 

Example  15a.  A  closed  tank  containing  100  cu.  ft.  of  air 
at  atmospheric  pressure  (=  14.5  Ibs.  per  sq.  in.)  is  to  be 
exhausted  down  to  5  bs.  by  a  pump  making  1  cu.  ft.  per 
stroke.  How  many  strokes  required  ? 

Solution.          ^  =  T~      and      -I-  =  152. 
pQ       14.5  V  +  v      101 

log  5  =  0.69897  log  100  =  2.00000 

log  14.5  =    1.16136  log  101  =  2.00432 

1.53761  1.99568 

These  two  logarithms  may  be  written  thus: 

-  1  +  0.53761  =  -  .46239        ,  .46239  =  10-  = 
-  1  +  0.99568  =  -  .00432  .00432 

If  the  volumetric  efficiency  of  the  machine  be  E,  then  the 
number  of  strokes  would  be  107  -f-  E. 


24  COMPRESSED  AIR 

The  results  found  under  Arts.  14  and  15  serve  well  to 
illustrate  the  curious  mathematical  gymnastics  that  com- 
pressed air  is  subject  to,  and  should  encourage  the  investi- 
gator who  likes  such  work,  and  should  put  the  designer  on 
guard. 

Art.  16.   Efficiency  when  Air  is  Used  without  Expansion. 

In  many  applications  of  compressed  air  convenience  and 
safety  are  the  prime  requisites,  so  that  power  efficiency 
receives  little  attention  at  the  place  of  application.  This 
is  so  with  such  apparatus  as  rock  drills,  pneumatic  hammers, 
air  hoists  and  the  like.  The  economy  of  such  devices  is  so 
great  in  replacing  human  labor  that  the  cost  in  power  is 
little  thought  of.  Further,  the  necessity  of  simplicity  and 
portability  in  such  apparatus  would  bar  the  complications 
needed  to  use  the  air  expansively.  There  are  other  cases, 
however,  notably  in  pumping  engines  and  devices  of  various 
kinds,  where  the  plant  is  fixed,  the  consumption  of  air  con- 
siderable and  the  work  continuous,  where  neglect  to  work 
the  air  expansively  may  not  be  justified. 

In  any  case  the  designer  or  purchaser  of  a  compressed-air 
plant  should  know  what  is  the  sacrifice  for  simplicity  or  low 
first  cost  when  the  proposition  is  to  use  the  air  at  full  pres- 
sure throughout  the  stroke  and  then  exhaust  the  cylinder 
full  of  compressed  air. 

Let  p  be  the  absolute  pressure  on  the  driving  side  of  the 
piston  and  pa  be  that  of  the  atmosphere  on  the  side  next 
the  exhaust.  Then  the  effective  pressure  is  p  —  pa  and  the 
effective  work  is  (p  —  pa)  v,  while  the  least  possible  work 
required  to  produce  this  air  is  pv  loge  r. 

Hence  the  efficiency  is  E  =  ^  ~  Pa}  v  - 

pv  loge  r 

Dividing  numerator  and  denominator  by  pav  this  reduces  to 

E=±=±*  (15) 

r  loge  r 

This  is  the  absolute  limit  to  the  efficiency  when  air  is  used 
without  expansion  and  without  reheating.  It  cannot  be 
reached  in  practice. 

Table  VI  represents  this  formula.     Note  that  the   effi- 


FORMULAS   FOR  WORK  25 

ciency  decreases  as  r  increases.  Hence  it  may  be  justi- 
fiable to  use  low-pressure  air  without  expansion  when  it 
would  not  be  if  the  air  must  be  used  at  high  pressure. 

Clearance  in  a  machine  of  t  this  kind  is  just  that  much 
compressed  air  wasted.  If  clearance  be  considered,  Eq.  (15) 
becomes 

E  =  ~ 


.1_L. 
(l+c)rloger 

where  c  is  the  percentage  of  clearance.     In  some  machines, 
if  this  loss  were  a  visible  leak,  it  would  not  be  tolerated. 

Art.  17.  Variation  of  Atmospheric  Pressure  with  Altitude. 

In  most  of  the  formulas  relating  to  compressed-air  opera- 
tions the  pressure  pa,  or  weight  wa,  of  free  air  is  a  factor. 
This  factor  varies  slightly  at  any  fixed  place,  as  indicated 
by  barometer  readings,  and  it  varies  materially  with  varying 
elevations. 

To  be  precise  in  computations  of  work  or  of  weights  or 
volumes  of  air  moved,  the  factors  pa  and  wa  should  be  deter- 
mined for  each  experiment  or  test,  but  such  precision  is 
seldom  warranted  further  than  to  get  the  value  of  pa  for 
the  particular  locality  for  ordinary  atmospheric  conditions. 
This  however  should  always  be  done.  It  is  a  simple  matter 
and  does  not  increase  the  labor  of  computation.  In  many 
plants  in  the  elevated  region  pa  may  be  less  than  14.0  Ibs. 
per  sq.  in.,  and  to  assume  it  14.7  would  involve  an  error  of 
more  than  5%. 

Direct  reading  of  a  barometer  is  the  easiest  and  usual  way 
of  getting  atmospheric  pressure,  but  barometers  of  the 
aneroid  class  should  be  used  with  caution.  Some  are 
found  quite  reliable,  but  others  are  not.  In  any  case  they 
should  be  checked  by  comparison  with  a  mercurial  barom- 
eter as  frequently  as  possible. 

If  m  be  the  barometer  reading  in  inches  of  mercury  and 
F  be  the  temperature  (Fahrenheit),  the  pressure  in  pounds 
per  sq.  in.  is 


oU 
.4931  m  [1  -  .0001  (F  -  32)].  (10) 


26  COMPRESSED  AIR 

The  information  in  Table  II  will  usually  obviate  the  need  of 
using  Eq.  (16). 

In  case  the  elevation  is  known  and  no  barometer  available 
the  problem  can  be  solved  as  follows: 

Let  ps  =  pressure  of  air  at  sea  level, 
ws  —  weight  of  air  at  sea  level, 
px,  wx  be  like  quantities  for  any  other  elevation. 

Then  in  any  vertical  prism  of  unit  area  and  height  dh  we 
have 

Px  +  dpx  =  px  +  wxdh, 

or  dpx  =  wxdh. 

But  1^  =  2*.  therefore  dpx  =  —  pxdh, 

w8       ps  ps 

or  dh  =  &  &*,  and  therefrom  h  =  ^  X  log  2t, 

Ws     Px  Ws  pa 

where  pa  is  the  pressure  at  elevation  h  above  sea  level.     Sub- 
stitute for  ws  its  equivalent 

h  v, 

ve  get  7^—^-  =  log  ^  . 
53.17 1  pa 

Whence  loge  pa  =  loge  ps  -  ^  ^  ^ 

Making  ps  =  14.745  and  adopting   to    common  logarithm 
and  Fahrenheit  temperatures, 

login  Pa  =  1.16866  -  .„  .  ,j  .    .       •  (17) 


Table  V  is  made  up  by  formula  17. 


CHAPTER  II 
A 

MEASUREMENT  OF  Am 

Art.  18.     General  Discussion. 

Progress  in  the  science  of  compressed-air  production  and 
application  has  evidently  been  hindered  by  a  lack  of  accu- 
rate data  as  to  the  amount  of  compressed  air  produced  and 
used. 

The  custom  is  almost  universal  of  basing  computations 
on,  and  of  recording  results  as  based  on,  catalog  rating  of 
compressor  volumes  —  that  is,  on  piston  displacement. 

The  evil  would  not  be  so  great  if  all  compressors  had 
about  the  same  volumetric  efficiency,  but  it  is  a  fact  that  the 
volumetric  efficiency  varies  from  60  per  cent  to  90  per  cent, 
depending  on  the  make,  size,  condition  and  speed  of  the 
machine;  no  wonder,  then,  that  calculations  often  go  wrong 
and  results  seem  to  be  inconsistent. 

There  are  problems  in  compressed-air  transmission  and  use 
for  the  solution  of  which  accurate  knowledge  of  the  volume 
or  weight  of  air  passing  is  absolutely  necessary.  Chief  among 
these  are  the  determination  of  friction  factors  in  air  pipes 
and  the  efficiency  of  pumps,  including  air  lifts. 

Purchasers  may  be  imposed  upon,  and  no  doubt  often  are, 
in  the  purchase  of  compressors  with  abnormally  low  volu- 
metric efficiencies.  Contracts  for  important  air-compressor 
installation  should  set  a  minimum  limit  for  the  volumetric 
efficiency,  and  the  ordinary  mechanical  engineer  should  have 
knowledge  and  means  sufficient'  to  test  the  plant  when 
installed. 

There  is  little  difficulty  in  the  measurement  of  air.  The 
calculations  are  a  little  more  technical,  but  the  apparatus 
is  as  simple  and  the  work  much  less  disagreeable  than  in 
measurements  of  water. 

In  many  text-books  theoretic  formulas  are  presented  for 

27 


28 


COMPRESSED  AIR 


the  flow  of  air  at  high  pressures  through  orifices  into  the 
atmosphere.  Such  formulas  are  complicated  by  the  neces- 
sity of  considering  change  of  volume  and  temperature,  and 
even  where  the  proper  empirical  coefficients  are  found  the 
formulas  are  unwieldy. 

Art.  19.     Apparatus  for  Measuring  Air  by  Orifice. 

Present  indications  are  that  the  standard  method  of 
determining  flow  of  air  will  require  the  pressure  to  be  reduced 
to  less  than  one  foot  head  of  water  in  order  that  change 
of  volume  and  temperature  may  be  neglected  and  the  for- 
mula simplified  thereby. 

Experiments  under  such  circumstances  show  coefficients 
even  more  constant  than  those  for  standard  orifices  for 
measuring  water.  The  coefficients  given  in  Art.  20  were 
determined  at  McGill  University  by  methods  and  apparatus 
described  first  in  Trans.  Am.  So.  Mech.  E.,  Vol.  27,  Dec.,  1905, 
and  later  in  Compressed  Air,  Sept.,  1906,  p.  4187. 

Having  the  coefficients  once  determined,  the  necessary 
apparatus  is  simple  and  inexpensive.  The  essentials  are 
shown  in  Fig.  7. 


,'B 

> 

I 

J 

b 

b 

b 

] 

1                D 

j 

ri? 

Fig.  7. 

A  —  Compressed-air  pipe, 

B  =  Closed  box  or  cylinder. 

T  =  Throttle, 

b  =  Baffle  boards  or  screen, 

H  =  Thermometer, 

C  =  Cork, 

0  =  Orifice  in  thin  metal  plate  (Standard), 

U  =  Bent  glass  tube  containing  colored  water, 

G  =  Scale  of  inches. 


MEASUREMENT  OF  AIR  29 

The  box  B  may  be  mads  of  any  convenient  light  material. 
The  pressure  is  only  a  few  ounces  and  the  tendency  to  leak 
slight.  The  purpose  of  the  throttle  T  is  to  control  the 
pressure  against  which  the  compressor  works.  The  appro- 
priate orifice  0  can  be  determined  by  a  preliminary  com- 
putation, assuming  i  at'say  5".  See  Eq.  (18). 

In  testing  a  compressor  it  should  be  run  until  every  part 
is  at  its  normal  running  temperature.  By  means  of  the 
throttle  T  the  compressor  can  be  worked  under  various 
pressures  and  speed  and  thereby  its  individual  curves  of 
volumetric  efficiency  obtained. 

Art.  20.  Formula  for  Standard  Orifice  under  Low 
Pressure. 

Let  pa  =  air  pressure  in  Ibs.  per  sq.  in.  inside  the  box, 
Q  =  weight  of  air  passing  per  second, 
wa  =  weight  of  a  cubic  foot  of  air  in  pounds, 
d  =  diameter  of  orifice  in  inches, 
i  =  pressure  as  read  on  water  gage  in  inches, 
t  —  absolute  temperature  Fahrenheit's  scale,  inside 

box, 

c  =  the  experimental  coefficient. 

Where  changes  of  density  and   temperature  can   be   neg- 
lected the  theoretic  velocity  through  the  orifice  is  v  =  ^2  gh 
where  h  is  the  head  of  air  of  uniform  density  that  would 
produce  the  pressure  head  i. 

Hence      h  =  -^  X  — ;  therefore    v  =  \l 2  g  ±  X  —  • 
12        wa  12        wa 

70 

But  Q  =  waXav  where  a  =  area  of  orifice  in  sq.  ft.  =  — - — 

4  X  144 

Inserting  these  values  and  putting  wa  under  the  radical  there 
results 

o  =       ird2      .  L  .  i     62.5 
4X144 

But  Wa  =  53.17 1 


Therefore      Q  =  .0136d2  J\  par  =  .1632  d2  J\  pa  (18) 

»  t  *  t 

where  pa  is  in  Ibs.  per  sq.  in. 


30  COMPRESSED  AIR 

The  pressure  due  to  i  (  =  .036  i)  should  be  included  in  pa. 
If  the  work  is  at  sea  level  and  pressure  i  be  neglected, 
pa=  14.7  X  144  and  the  formula  becomes 


Q  =  .6299  d2^-,  (18a) 

which  is  the  formula  published  by  McGill  University. 

This  is  the  theoretic  formula.  To  it  must  be  applied  the 
experimental  coefficient  c  as  given  in  Table  VIII.  Note 
that  c  varies  but  little  from  0.60,  and  the  same  c  can  be 
used  in  Eq.  (18)  and  (18a). 

Example  20a.  In  a  run  with  the  apparatus  shown  in 
Fig.  7  the  following  was  recorded:  d  =  2.32";  i  =  4.6"; 
T  =  186°  F.  inside  drum,  T=  86°  F.  in  free  air.  Elevation 
1200'.  Find  the  weight  and  volume  of  free  air  passing. 

Solution.  From  Table  II,  interpolating  for  86°  in  the  line 
with  1200  elevation  we  get  wa  =.0700  and  pa  for  free  air 
=  14.1.  Add  the  pressure  due  to  i  (  =  .036  X  4.6)  and  we 
get  the  corrected  pa  =  14.26.  In  Table  VIII  the  coefficient 
for  d  =  2.32  and  i  =  4.6  is  0.599.  These  numbers  inserted 
in  (18)  give 

Q  =  .599  X  .1632  X  (2.32)2l/~^  X  14.26 

T  64o 

=  .  1684  pound  per  second 
and  the  free  air  volume 

X  60  =  144.3  cu.  ft.  per  minute. 


.0700 
ByEq.  (18a)  Q=  .1747. 

Art.  21.   Air  Measurement  in  Tanks. 

The  amount  of  air  put  into  or  taken  out  of  a  closed  tank 
or  system  of  tanks  and  pipes,  of  known  volume,  can  be 
accurately  determined  by  Eq.  (3),  viz., 


PxVx         tx  pa     tx 

By  this  means  the  volume  of  air  delivered  into  a  closed  sys- 
tem by  a  compressor  can  be  very  accurately  determined. 

The  process  would  be  as  follows:  Determine  the  volumes 
of  all  tanks,  pipes,  etc.,  to  be  included  in  the  closed  system, 


MEASUREMENT  OF  AIR  31 

open  all  to  free  air  and  observe  the  free-air  temperature; 
then  switch  the  delivery  from  the  compressor  into  the 
closed  system;  count  the  strokes  of  the  compressor  until  the 
pressure  is  as  high  as  desired;  then  shut  off  the  closed  tank 
and  note  pressure  and  temperatures  of  each  separate  part 
of  the  volume.  Then  the  formula  above  will  give  the  vol- 
ume of  free  air  which  compressed  and  heated  would  occupy 
the  tanks.  From  this  subtract  the  volume  of  free  air  origi- 
nally in  the  tanks;  the  remainder  will  be  what  the  compressor 
has  delivered  into  the  system.  Note  that  the  compressor 
should  be  running  hot  and  at  normal  speed  and  pressure 
when  the  test  is  made  for  its  volumetric  efficiency. 

Usually  the  temperature  changes  will  be  considerable,  but 
if  the  system  is  tight,  time  can  be  given  for  the  temperature 
to  come  back  to  that  of  the  atmosphere,  thus  saving  the 
necessity  of  any  temperature  observations. 

Where  a  convenient  closed-tank  system  is  available  this 
method  is  recommended. 

This  method  —  that  is,  Eq.  (3)  as  stated  above  —  was  used 
to  determine  the  quantity  of  air  passing  the  orifices  in  the 
experiments  by  which  the  coefficients  were  determined  as 
given  in  Art.  20,  Table-VII. 

Example  21a.  A  tank  system  consists  of  one  receiver  3' 
diam.  X  12',  one  air  pipe  6"  X  40',  one  4"  X  4000/  and  a 
second  receiver  at  end  of  pipe  2'  diam.  X  8'.  A  compres- 
sor 12"  X  18"  with  li"  piston  rod  puts  the  air  from  1250 
revolutions  into  the  system,  after  which  the  pressure  is 
80  gage  and  temperature  in  first  receiver  200°,  while  in 
other  parts  of  the  tank  system  it  is  60°.  Temperature  of 
outside  air  being  50°,  pa  =  14.5  per  sq.  in.  Find  volu- 
metric efficiency  of  the  compressor. 

Solution.   Volumes  (from  Table  X) : 

1st  receiver     84.84  cu.  ft." 
6"  pipe  7.84 


I"  pipe  349.20 


382.16 


2nd  receiver     25.12 
Total  467.00  in  tank  system. 


32  COMPRESSED  AIR 

Piston   displacement   in   one   revolution  =  2.338   cu.   ft. 
(piston  rod  deducted). 


By  formula  va  =       -^j  X  -note  that  the  quantity  in  paren- 

\  Pa  /          lx 

thesis  is  constant  and  therefore  a  slide  rule  can  be  conven- 
iently used,  otherwise  work  by  logarithms. 


,„  „„,„«„„  .  .  x  __  .  »„ 


va  in  6"  pipe,  4"  pipe  and  second  receiver  with  total 

volume  382.16  and  i  =  60°  =  .  2447.1 


Total 2864.3 

Original  volume  of  free  air 467 


Volume  of  free  air  added 2397.3 

2397.3  -r-  2.338  =  1028. 
Therefore  the  volumetric  efficiency  is 

E  =  1028  ^  1250  =  82%. 


CHAPTER*  III 

FRICTION  IN  AIR  PIPES 

Art.  22.  In  the  literature  on  compressed  air  many  for- 
mulas can  be  found  that  are  intended  to  give  the  friction  in 
air  pipes  in  some  form.  Some  of  these  formulas  are,  by 
evidence  on  their  face,  unreliable,  as  for  instance  when  no 
density  factor  appears ;  the  origin  of  others  cannot  be  traced 
and  others  are  in  inconvenient  form.  Tables  claiming  to 
give  friction  loss  in  air  pipes  are  conflicting,  and  reliable 
experimental  data  relating  to  the  subject  are  quite  limited. 

In  this  article  and  the  next  are  presented  the  derivation 
of  rational  formulas  for  friction  in  air  pipes  with  full  exposi- 
tion of  the  assumptions  on  which  they  are  based.  The  coeffi- 
cients were  gotten  from  the  data  collected  in  Appendix  B. 

Art.  23.   The  Formula  for  Practice. 

The  first  investigation  will  be  based  on  the  assumption  that 
volume,  density  and  temperature  remain  constant  through- 
out the  pipe. 

Evidently  these  assumptions  are  never  correct;  for  any 
decrease  in  pressure  is  accompanied  by  a  corresponding 
increase  in  volume  even  if  temperature  is  constant.  (The 
assumption  of  constant  temperature  is  always  permissible.) 
However,  it  is  believed  that  the  error  involved  in  these 
assumptions  will  be  less  than  other  unavoidable  inaccuracies 
involved  in  such  computations. 

Let  /  =  lost  pressure  in  pounds  per  sq.  in., 
I  =  length  of  pipe  in  feet, 
d  =  diameter  of  pipe  in  inches, 
s  =  velocity  of  air  in  pipe  in  feet  per  second, 
r  =  ratio  of  compression  in  atmospheres, 
c  =  an  empirical  coefficient  including  all  constants. 
33 


34  COMPRESSED  AIR 

Experiments  have  proved  that  fluid  friction  varies  very 
nearly  with  the  square  of  the  velocity  and  directly  with 
the  density.  Hence  if  k  be  the  force  in  pounds  necessary 
to  force  atmospheric  air  (r  =  1)  over  one  square  foot  of  sur- 
face at  a  velocity  of  one  foot  per  second,  then  at  any  other 
velocity  and  ratio  of  compression  the  force  will  be 


and  the  force  necessary  to  force  the  air  over   the  whole 
interior  of  a  pipe  will  be 


and  the  work  done  per  second,  being  force  multiplied  by 
distance,  is 

Work  =  ^XArs3. 

Now  if  the  pressure  at  entrance  to  the  pipe  is  /  pounds  per 
sq.  in.  greater  than  at  the  other  end,  the  work  per  second 
due  to  this  difference  (neglecting  work  of  expansion  in  air)  is 

Work  =  /  —  s. 
4 

Equating  these  two  expressions  for  work  there  results 


or 


Now  the  volume  of  compressed  air,  v,  passing  through  the 
pipe  is,  in  cubic  feet, 

ird2 

v  =   --  s 
4X  144 

and  the  volume  of  free  air,  va,  is  rv. 

Therefore  va  =      ****      X  rs 

4  X  144 

*-< 


FRICTION  IN  AIR  PIPES 


35 


Insert  this  value  of  s2  in  Eq.  (19)  and  reduce  and  there  results 
,      4   ,  /4  X  144V   I  v 


or 


=c. 
C  d*   r 


(20) 


where  c  is   the  experimental   coefficient  and  includes  all 
constants. 


From  Eq.  (20), 


/jcWV 
\fr  I 


(21) 


From  the  data  collected  in  Appendix  B  the  following 
results  were  computed.  In  this  r  and  s  are  mean  results 
and  c  is  the  average  of  all  the  runs  made  on  each  pipe. 


d 

c 

r 

s 

t 

1 

.092 

2.4to    8.0 

29  to    70 

60°  F. 

* 

.076 

1.5to  10.2 

35  to  100 

100 

.084 

l.Sto  10.8 

10  to    50 

80 

2 

.080 

2.0to  10.6 

5  to    28 

80 

3 

.072 

4 

12  to  100 

60 

4 

.066 

7 

28 

35 

5 

.057 

5 

30 

86 

6 

.066 

4.5 

33 

70 

8 

.061 

4.5 

20 

70 

12 

.047 

7.5 

20 



An  examination  of  the  data  in  Appendix  B  shows  that  the 
coefficient  c  is  independent  of  r  and  of  s.  If  it  is  affected  by 
the  temperature  it  cannot  be  detected  in  these  data.  In 
relation  to  the  diameters  c  evidently  increases  as  the  diam- 
eter decreases.  A  plot  of  diameters  and  c  on  coordinate 
paper  gives  a  straight  line  and  reveals  the  relation  c  = 
.0866  —  .0033  d  as  most  nearly  averaging  the  results.  This 
gives  the  following  values  for  c*: 


Diameters        *        1       1J      2        2£       3        4        5        6        8       10      12 
Coefficients^. 085  .083  .081  .080  .079  .078  .073  .070  .067  .060  .053  .047 


Formulas  (20)  and  (21)  would  be  theoretically  a  little  more 

accurate  if  va  were  expressed  in  terms  of  the  actual  weight 

of  air  passing.     This  would  involve  the  observed  free  air 

pressure  and  temperature  at  the  time  considered.      Such  a 

*  See  Appendix  C,  page  122. 


36  COMPRESSED  AIR 

modification  renders  the  formula  much  more  laborious 
and  would  probably  add  nothing  to  its  value  for  practical 
purposes. 

Table  IX  and  Plates  0,  I,  II,  III,  and  IV  are  based  on 
formula  (20). 

Art.  24.    Theoretically  Correct  Friction  Formula. 

The  theoretically  correct  formula  for  friction  in  air 
pipes  must  involve  the  work  done  in  expansion  while  the 
pressure  is  dropping. 

Let  pi  and  p2  be  the  absolute  pressures  at  entrance  and 
discharge  of  the  pipe  respectively  and  let  Q  be  the  total 
weight  of  air  passing  per  second. 

Then  the  total  energy  in  the  air  at  entrance  is 


and  at  discharge,  the  energy  is 


Pa      2g 

The  difference  in  these  two  values  must  have  been  absorbed 
in  friction  in  the  pipe.  Hence,  using  the  expression  for 
work  done  in  friction  that  was  derived  in  Art.  23,  we  get 

-3  2  « 


*J   IkTS*    =    pava  (log  Ei   -  log  2l) 
12  V          Pa  Pal 


+ 


Numerical   computations  will  show  the  last  term,  viz. 

-*-  (s22  —  Si2)  is  relatively  so  small  that  it  can  be  neglected  in 
^  Q 

any  case  in  practice  without  appreciable  error.     Hence  by  a 
simple  reduction  we  get 


X  but 


p2      12  pa        va  4 

which  when  substituted  gives 


_I  =  x 

d 


or  considering  pa  as  constant, 

Iog102l=  c    *s» 
Pz  d 

or  logio  p2  =  logic,  Pi  -  ci  3§2.  (22) 

a 


FRICTION  IN  AIR  PIPES  37 

In  Eq.  (22)  Ci  is  the  experimental  coefficient  and  includes 
all  constants,  s  is  the  velocity  in  the  air  pipe  and  varies 
slightly  increasing  as  the  pressure  drops.  All  efforts  so  far 
have  failed  to  get  a  formula  in  satisfactory  shape  that  makes 

allowance  for  the  variation  in  s.  - 

i 

In  working  out  Ci  from  experimental  data  s  should  be  the 
mean  between  the  Si  and  s2,  and  when  using  the  formula 
the  s  may  be  taken  as  about  5  per  cent  greater  than  Si. 

Note  that  in  the  solution  of  Eq.  (22)  common  logarithms 
should  be  used  for  convenience,  allowing  the  modulus,  2.3+, 
to  go  into  the  constant  c\. 

The  working  formula  may  be  put  in  a  different  and 
possibly  a  more  convenient  form,  thus.  In  the  expression 

,       £i  =  2L*xjB_rs, 

p2      12       pava 

substitute  for  s  its  value 

4  X  144  va 


s  — 


ird-r 
and  reduce  and  we  get 

(23) 


Still  another  form  is  gotten  thus.      The  whole  weight  of  air 
passing  is  va  X  wa  =  Q,  and  by  Eq.  (12) 

Q  =  va  -~^ —   and  therefore  va  =  — : —  • 

53. 17*  pa 

Also  rx  =  ^    and  wa=  -&- - . 

pa  53.17£ 

Substitute  these  in  (23)  and  it  reduces  to 

log  p2  =  log  Pl  -  c2  -^L  ( 2-Y  (24) 

Wad5  \px/ 

In  ordinary  practice  —  may  be  taken  as  constant.     If  this 
be  done  Eq.  (24)  becomes 

log  pz  =  log  pi  -  c3  -  r*-\  -  (24a) 

If  ta  =  525  and  wa  =  .075,  then  c3  =  7000  o*. 


38  COMPRESSED  AIR 

In  (24)  and  (24a)  px  varies  between  pi  and  p2.  Careful 
computations  by  sections  of  a  long  pipe  show  px  to  vary  as 
ordinates  to  a  straight  line.  Modifying  the  formulas  to 
allow  for  this  variation  renders  them  unmanageable.  In 
working  out  the  coefficient  px  may  be  taken  as  a  mean 
between  pi  and  p2,  and  in  using  the  formula  p  may  be  taken 
as  pi  less  half  of  the  assumed  loss  of  pressure. 

As  before  suggested,  common  logarithms  should  be  used  in 
all  the  equations  of  this  article. 

Finally  it  should  be  said  that  extreme  refinement  in  com- 
puting friction  in  air  pipes  is  a  waste  of  labor,  for  there  are 
too  many  variables  in  practical  conditions  to  warrant  much 
effort  at  precision. 

A  study  of  the  data  collected  in  Appendix  B  gives 
values  for  c2,  Eq.  (24),  that,  for  pipes  three  to  twelve  inches 
diameter,  conform  closely  to  the  expression 

c2  =  .0124  -  .0004  d, 
which  gives  the  following: 

d"  =      3 
C2  =  .0112 

C3  =    78.4 

With  these  coefficients  px  in  equations  (24)  and  (24a)  is  to 
be  taken  in  pounds  per  square  inch. 

Equations  (24)  and  (24a)  are  theoretically  more  correct 
than  Eq.  (20)  and  the  coefficients  of  the  former  will  not  vary 
so  much  as  those  for  the  latter,  but  when  the  coefficients  are 
correctly  determined  for  Eq.  (20)  it  is  much  easier  to  com- 
pute and  can  be  adapted  to  tabulation,  while  Eq.  (24)  can- 
not be  tabulated  in  any  simple  way. 

Example  24a.  Apply  formulas  (20)  and  (24)  to  find  the 
pressure  lost  in  1000'  of  4"  pipe  when  transmitting  1200 
cu.  ft.  free  air  per  minute  compressed  to  150  gage  when  at- 
mospheric conditions  are  pa  =  14.0,  wa  =  .  073  and  ta  =  540. 

Solution  by  Eq.  (20) :    r  =  15Q+  14  =  11.71.    By  Table  IX 

divide  23.44  by  11.71  and  the  result,  2  pounds,  is  the  pres- 
sure lost  per  1000'. 


4 

5 

6 

8 

10 

12 

.0108 

.0104 

.0100 

.0092 

.0084 

.0080 

75.6 

72.8 

70.0 

64.4 

58.8 

56.0 

FRICTION   IN  AIR  PIPES 


39 


Solution  ofEq.  (24)  :   The  coefficient  for  a  4"  pipe  is  .0108, 
and  log  pi  =  log  (150  +  14)  =  2.214844. 


Then  log  P2  -  ,214844  -  .0108 


X 


The  log  of  the  last  term  is  3.791193  and  its  corresponding 
number  is  .006183. 

2.214844  -  .0(16183  =  2.208661  =  log  p2. 
Whence  p2  =  161.7+     and     pi  -  p2  =  2.3. 

Art.  25.  Efficiency  of  Power  Transmission  by  Compressed 
Air. 

In  the  study  of  propositions  to  transmit  power  by  piping 
compressed  air,  persons  unfamiliar  with  the  technicalities 
of  compressed  air  are  apt  to  make  the  error  of  assuming 
that  the  loss  of  power  is  proportional  to  the  loss  of  pressure, 
as  is  the  case  in  transmitting  power  by  piping  water.  Fol- 
lowing is  the  mathematical  presentment  of  the  subject: 

Pi  =  absolute  air  pressure  at  entrance  to  transmission  pipe, 
p2  =  absolute  air  pressure  at  end  of  transmission  pipe, 
i>i  =  volume  of  compressed  air  entering  pipe  at  pressure  pi, 
v2  =  volume  of  compressed  air  discharged  from  pipe  at 
pressure  p2. 

Then   crediting   the   air  with   all   the   energy   it   can   de- 
velop  in   isothermal    expansion,    the    energy    at   entrance 

log  n,  and  at  discharge  the  energy  is 


s 


log        = 

Pa 


g^  =  p«#2  Iogr2. 

^° 

Hence  efficiency  E  = 


loge 


~»N 

=  jog.*  . 
loge  ri 


(25) 


Common  logs  may  be  used  since  the  modulus  cancels.     The 
varying  efficiencies  are  illustrated  by  the  following  tables. 

pa=  14.5.     pi  =  87.     ri  =  6.     log  n  =  .7781. 


P2  

85 

80 

75 

70 

65 

60 

r2  

5.86 

5.52 

5.17 

4.83 

4.48 

4.14 

Iogr2.  . 

.7679 

.7419 

.7135 

.6839 

.6513 

.6170 

E  

.987 

.953 

.917 

.879 

.837 

.793 

40 


COMPRESSED  AIR 
=  14.5.     pi  =  145.    n  =  10.    log  n  =  1.000. 


T>«  • 

140 

135 

130 

125 

120 

r 

9  66 

9  31 

8  97 

8  62 

8  28 

,2  

losrr, 

9850 

9689 

9528 

9355 

9185 

£  2 
iii  

98 

97 

95 

93 

92 

The  above  examples  illustrate  the  advantage  in  trans- 
mitting at  high  pressure.  Of  course  the  work  cannot  be 
fully  recovered  in  either  case  without  expanding  down  to 
atmospheric  pressure,  and  to  do  this  in  practice  heating 
would  be  necessary.  It  should  be  understood  also  that  by 
reheating  this  efficiency  can  be  exceeded. 

It  should  be  noted  also  that  the  above  does  not  apply 
in  cases  where  the  air  is  applied  without  expansion.  In 
such  cases  the  efficiency  of  transmission  alone  would  be 

E=    (P2  ~  Pa)  V2  =  n  (>2   ~    1)  < 
(Pi  -  Pa)  Vi         r2  (ri  -   1) 

Example  25a.  What  diameter  of  pipe  will  transmit  5000 
cu.  ft.  of  free  air  per  minute  compressed  to  100  Ibs.  gage,  with 
a  loss  of  10  per  cent  of  its  energy  in  2500  feet  of  pipe,  assum- 
ing pa  =  14.0? 

Solution. 


Whence  log  r2  =  0.8200;  r2  =  6.6,  and  6.6  X  14  =  92.4. 
/  =  114  -  92.4  =  21.6  =  loss  of  pressure. 
By  Eq.  (21), 


log  d  = 


=  |[log  ( 


.06X  2500)  X 


log  ( 21.6  X 


8.15  +  6.( 


=  .7602,  whence  d  =  5.75"., 


Otherwise  go  into  Table  IX  with  loss  for  1000  ft.  = 

.2.0 

=  8.64,  and  8.64  X  r  =  8.64  X  7.37  =  63,  (7.37  being  the 
meanr).  Then  opposite  5000  in  the  first  column  find  nearest 
value  to  63,  which  is  55  in  the  6"  column;  showing  the  re- 
quired pipe  to  be  a  little  less  than  6". 


CHAPTER   IV 

.• 
OTHER  ~  AIR   COMPRESSORS 

Art.  26.    Hydraulic  Air  Compressors. — Displacement  Type. 

Compressors  of  this  type  are  of  limited  capacity  and  low 
efficiency,  as  will  be  shown.  They  are  therefore  of  little 
practical  importance.  However,  since  they  are  frequently 
the  subject  of  patents  and  special  forms  are  on  the  market, 
their  limitations  are  here  shown  for  the  benefit  of  those  who 
may  be  interested. 

Omitting  all  reference  to  the  special  mechanisms  by  which 
the  valves  are  operated,  the  scheme  for  such  compressors  is 
to  admit  water  under  pressure  into  a  tank  in  which  air  has 
been  trapped  by  the  valve  mechanisms.  The  entering 
water  brings  the  air  to  a  pressure  equal  to  that  of  the  water; 
after  which  the  air  is  discharged  to  the  receiver,  or  point  of 
use.  When  the  air  is  all  out  the  tank  is  full  of  water,  at 
which  time  the  water  discharge  valves  open,  allowing  the 
water  to  escape  and  free  air  to  enter  the  tank  again,  after 
which  the  operation  is  repeated.  Usually  these  operations 
are  automatic.  The  efficiency  of  such  compression  is  limited 
by  the  following  conditions. 

Let  P  =  pressure  of  water  above  atmosphere,  or  ordinary 

gage  pressure, 
V  =  volume  of  the  tank. 

Then  P  +  pa  =  absolute  pressure  of  air  when  compressed. 
The  energy  represented  by  one  tank  full  of  water  is  PV  and 
by  one  tank  full  of  free  air  when  compressed  to  P  +  pa  is 

P  4-  v 
paV  loge  -   I-^  =  paV  log,  r. 

Pa 

Therefore  the  limit  of  the  efficiency  is 
™  _  paV  loger  _  pa  loger 
PV  P 

41 


42  COMPRESSED  AIR 

But  P  =  pi  —  pa)  where  pi  is  the  absolute  pressure  of  the 
compressed  air.  Inserting  this  and  dividing  by  pa  the  expres- 
sion becomes 

E  =  loger  =  Iog10r  X  2.3  ^  ,26, 

T  —  1  T  —  1 

Table  VII  is  made  up  from  formula  (26). 

The  practical  necessity  of  low  velocities  for  the  water 
entering  and  leaving  the  tanks  renders  the  capacity  of  such 
compressors  low  in  addition  to  their  low  efficiency. 

Should  the  problem  arise  of  designing  a  large  compressor 
of  this  class  an  interesting  problem  would  involve  the  time  of 
filling  and  emptying  the  tank  under  the  varying  pressure  and 
head.  Since  it  is  not  likely  to  arise  space  is  not  given  it. 

It  is  possible  to  increase  the  efficiency  of  this  style  of 
compressor  by  carrying  air  into  the  tank  with  the  water  by 
induced  current  or  Sprengle  pump  action  —  a  well-known 
principle  in  hydraulics.  At  the  beginning  of  the  action 
water  is  entering  the  tank  under  full  head  with  no  resist- 
ance, and  certainly  additional  air  could  be  taken  in  with  the 
water. 

Art.  27.    Hydraulic  Air  Compressors. — Entanglement  Type. 

A  few  compressors  of  this  type  have  been  built  compara- 
tively recently  and  have  proven  remarkably  successful  as 
regards  efficiency  and  economy  of  operation,  but  they  are 
limited  to  localities  where  a  waterfall  is  available,  and  the 
first  cost  of  installation  is  high. 

The  principle  involved  is  simply  the  reverse  of  the  air-lift 
pump,  and  what  theory  can  be  applied  will  be  found  in 
Art.  33  on  air-lift  pumps. 

Fig.  8  illustrates  the  elements  of  a  hydraulic  air  compressor. 
h  is  the  effective  water  fall. 
H  is  the  water  head  producing  the  pressure  in  the  compressed 

air. 

t  is  a  steel  tube  down  which  the  water  flows. 
S  is  a  shaft  in  the  rock  to  contain  the  tube  t  and  allow  the 

water  to  return. 

72  is  an  air-tight  hood  or  dome,  either  of  metal  or  of  natural 
rock,  in  which  the  air  has  time  to  separate  from  the  water. 


OTHER  AIR  COMPRESSORS 


43 


A  is  the  air  pipe  conducting  the  compressed  air  to  point 

of  use. 
b  is  a  number  of  small  tubes  open 

at  top   and  terminating  in    a 

throat   or   contraction,   in  the 

tube  t. 


By  a  well-known  hydraulic  prin- 
ciple, when  water  flows  freely  down 
the  tube  t  there  will  occur  suction  in 
the  contraction.  This  draws  air 
in  through  the  tubes  b,  which  air 
becomes  entangled  in  the  passing 
water  in  a  myriad  of  small  bub- 
bles; these  are  swept  down  with 
the  current  and  finally  liberated 
under  the  dome  R,  whence  the  air 
pipe  A  conducts  it  away  as  com- 
pressed air. 

The  variables  involved  practically 
defy  algebraic  manipulation,  so  that 
clear  comprehension  of  the  prin- 
ciples involved  must  be  the  guide 
to  the  proportions. 


Fig  8. 


Attention  to  the  following  facts  is  essential  to  an  intelli- 
gent design  of  such  a  compressor. 

1.  Air  must  be  admitted  freely  —  all  that  the  water  can 
entangle. 

2.  The  bubbles  must  be  as  small  as  possible. 

3.  The  velocity  of  the  descending  water  in  the  tube  t 
should  be  eight  or  ten  times  as  great  as  the  relative  ascend- 
ing velocity  of  the  bubble. 

The  ascending  velocity  of  the  bubble  relative  to  the  water 
increases  with  the  volume  of  the  bubble,  and  therefore 
varies  throughout  the  length  of  the  tube,  the  volume  of 
any  one  bubble  being  smaller  at  the  bottom  of  the  tube 
than  at  the  top.  For  this  reason  it  would  be  consistent  to 
make  the  lower  end  of  the  tube  t  smaller  than  the  top. 


44  COMPRESSED  AIR 

Efficiencies  as  high  as  80  per  cent  are  claimed  for  some 
of  these  compressors,  which  is  a  result  hardly  to  have  been 
expected. 

The  great  advantage  of  this  method  of  air  compression 
lies  in  its  low  cost  of  operation  and  in  its  continuity.  Me- 
chanical compressors  operated  by  the  water  power  could  be 
built  for  less  first  cost  and  probably  with  as  high  efficiency, 
but  cost  of  operation  would  be  much  higher. 

Art.  28.   Centrifugal  Air  Compressors. 

With  the  perfection  of  the  steam 'turbine  it  has  become 
practicable  to  deliver  air  at  several  atmospheres  pressure 
through  centrifugal  machines.  Such  machines  are  not  yet 
common,  but  doubtless  in  a  few  years  they  will  become  the 
standard  machine  where  large  volumes  of  air  are  needed  at 
low  and  constant  pressure.  The  simplicity,  compactness  and 
low  first  cost  of  such  machines  assure  them  a  popularity. 

The  theory  of  centrifugal  fans  or  air  compressors  would 
involve  matter  not  appropriate  to  the  purpose  of  this  vol- 
ume and  is  therefore  omitted. 

In  testing  centrifugal  compressors  or  blowers  the  orifice 
measurement,  Art.  20,  is  the  only  practicable  scheme.  If 
the  coefficients  have  not  been  determined  for  orifices  suffi- 
ciently large  to  pass  the  volume  of  air,  then  more  than  one 
orifice  can  be  placed  in  the  orifice  box.  It  is  not  necessary 
of  course  that  these  orifices  all  be  of  one  size. 

The  volume  of  air  delivered  and  the  efficiency  of  centrif- 
ugal fans  and  blowers  is  a  matter  little  understood,  seldom 
known,  and  often  far  from  what  is  assumed  or  claimed. 
The  remedy  for  this  is  to  be  found  in  intelligent  use  of  the 
orifice,  large  and  small;  and  for  such  purposes  the  deter- 
mination of  orifice  coefficients  such  as  shown  in  Table  V 
should  be  extended  to  orifices  all  the  way  up  to  two  feet  in 
diameter  in  order  to  test  very  large  ventilating  fans. 

Some  theoretic  discussion  of  centrifugal  fans  can  be  found 
in  Trans.  Am.  So.  C.  E.,  Vol.  51,  p.  12.  See  also  "  Turbo 
Compressors,"  Compressed  Air,  June,  1909,  p.  5364,  and  En- 
gineering Magazine,  Vol.  39,  p.  237. 


CHAPTER  V 

jit 

SPECIAL  APPLICATIONS  OF  COMPRESSED  AIK 

IN  this  chapter  attention  is  given  only  to  those  applications 
of  compressed  air  that  involve  technicalities  —  with  which 
the  designer  or  user  may  not  be  familiar,  or  by  the  discussion 
of  which  misuse  of  compressed  air  may  be  discouraged  and  a 
proper  use  encouraged. 

Art.  29.  The  Return-Air  System. 

In  the  effort  to  economize  in  the  use  of  compressed  air  by 
working  it  expansively  in  a  cylinder  the  designer  meets 
two  difficulties:  first,  the  machine  is  much  enlarged  when 
proportioned  for  expansion;  second,  it  is  considerably- more 
complicated;  and  third,  unless  reheating  is  applied  the  ex- 
pansion is  limited  by  danger  of  freezing. 

To  avoid  these  difficulties  it  has  been  proposed  to  use  the 
air  at  a  high  initial  pressure,  apply  it  in  the  engine  without  ex- 
pansion, and  exhaust  it  into  a  pipe,  returning  it  to  the  intake 
of  the  compressor  with  say  half  of  its  initial  pressure  remain- 
ing. The  diagram  Fig.  9  will  assist  in  comprehending  the 
system. 

To  illustrate  the  operation  and  theoretic  advantages  of 
the  system  assume  the  compressor  to  discharge  air  at  200 
pounds  pressure  and  receive  it  back  through  R  at  100 
pounds.  Then  the  ratio  of  compression  is  only  2  and  yet 
the  effective  pressure  in  the  engine  is  100  pounds. 

Evidently  then  with  a  ratio  of  compression  and  expansion 
of  only  2  the  trouble  and  loss  due  to  heating  are  practically 
removed;  and  the  efficiency  in  the  engine  even  without  a 
cut-off  would  be,  by  Eq.  (15)  72  per  cent.  By  the  above  dis- 
cussion the  advantages  of  the  system  are  apparent,  and  where 
a  compressor  is  to  run  a  single  machine,  as  for  instance  a 
pump,  the  advantage  of  this  return-air  system  will  surely 

45 


46 


COMPRESSED   AIR 


outweigh  the  disadvantage  of  two  pipes  and  the  high  pres- 
sure, but  where  one  compressor  installation  is  to  serve 
various  purposes  such  as  rock  drills,  pumps,  machine  shops, 
etc.,  the  system  cannot  be  applied.  There  should  be  a 
receiver  on  each  air  pipe  near  the  compressor. 


Fig.  9. 


Engine 


Art.  30.  The  Return-Air  Pumping  System. 

Following  the  preceding  article  it  is  appropriate  to  men- 
tion the  return-air  pumping  system.  The  economic  principle 
involved  is  different  from  that  of  the  return-air  system  in 
general. 

The  scheme  is  illustrated  in  Fig.  10.  It  consists  of  two 
tanks  near  the  source  of  water  supply.  Each  tank  is  con- 
nected with  the  compressor  by  a  single  air  pipe,  but  the  air 
pipes  pass  through  a  switch  whereby  the  connection  with 
the  discharge  and  intake  of  the  compressor  can  be  reversed, 
as  is  apparent  on  the  diagram.  In  operation,  the  compressor 


SPECIAL  APPLICATIONS   OF   COMPRESSED  AIR      47 

discharges  air  into  one  tank,  thereby  forcing  the  water  out 
while  it  is  exhausting  the  air  from  the  other  tanks,  thereby 
drawing  the  water  in.  The  charge  of  air  will  adjust  itself 
so  that  when  one  tank  is  emptied  the  other  will  be  filled, 
at  which  time  the  switch  will  automatically  reverse  the 
operation. 


=^5-  Water  Supply 


Fig.  10. 

The  economic  advantage  of  the  system  lies  in  the  fact  that 
the  expansive  energy  in  the  air  is  not  lost  as  in  the  ordinary 
displacement  pump  (Art.  31).  The  compressor  takes  in  air 
at  varying  degrees  of  compression  while  it  is  exhausting  the 
tank. 

The  mathematical  theory  and  derivation  of  formulas  for 
proportioning  this  style  of  pump  are  quite  complicated  but 
interesting.  Since  the  system  is  patented,  further  discus- 
sion would  seem  out  of  place.  It  will  be  found  in  Trans. 
Am.  So.  C.  E.,  Vol.  54,  p.  19. 


48  COMPRESSED  AIR 

Art.  31.  Simple  Displacement  Pump.  First  known  as 
the  Shone  ejector  pump. 

In  this  style  of  pump  the  tank  is  submerged  so  that  when 
the  air  escapes  it  will  fill  by  gravity.  The  operation  is  simply 
to  force  in  air  and  drive  the  water  out.  When  the  tank  is 
emptied  of  water,  a  float  mechanism  closes  the  compressed- 
air  inlet  and  opens  to*  the  atmosphere  an  outlet  through 
which  the  air  escapes,  allowing  the  tank  to  refill.  Various 
mechanisms  are  in  use  to  control  the  air  valve  automati- 
cally. The  chief  troubles  are  the  unreliable  nature  of  float 
mechanisms  and  the  liability  to  freezing  caused  by  the 
expansion  of  the  escaping  air.  Some  of  the  late  designs 
seem  reliable. 

The  limit  of  efficiency  of  this  pump  is  given  by  formula  15 
and  Table  VI.  The  pump  is  well  adapted  to  many  cases 
where  pumping  is  necessary  under  low  lifts.  In  case  of  drain- 
age of  shallow  mines  and  quarries,  lifting  sewerage,  and  the 
like,  one  compressor  can  operate  a  number  of  pumps  placed 
where  convenient ;  and  each  pump  will  automatically  stop 
when  the  tank  is  uncovered  and  start  again  when  the  tank  is 
again  submerged, 


CHAPTER  VI 

i 

AIR-LIF*  PUMP 


Art.  32.  The  air-lift  pump  was  introduced  in  a  practical 
way  about  1891,  though  it  had  been  known  previously,  as 
revealed  by  records  of  the  Patent  Office.  The  first  effort  at 
mathematical  analysis  appeared  in  the  Journal  of  the  Frank- 
lin Institute  in  July,  1895,  with  some  notes  on  patent  claims. 
In  1891  the  United  States  Patent  Office  twice  rejected  an 
application  for  a  patent  to  cover  the  pump  on  the  ground 
that  it  was  contrary  to  the  law  of  physics  and  therefore  would 
not  work.  Altogether  the  discovery  of  the  air-lift  pump 
served  to  show  that  at  that  late  date  all  the  tricks  of  air 
and  water  had  not  been  found  out. 

The  air  lift  is  an  important  addition  to  the  resources  of  the 
hydraulic  engineer.  By  it  a  greater  quantity  of  water  can 
be  gotten  out  of  a  small  deep  well  than  by  any  other  known 
means,  and  it  is  free  from  the  vexatious  and  expensive  depre- 
ciation and  breaks  incident  to  other  deep  well  pumps.  While 
the  efficiency  of  the  air  lift  is  low  it  is,  when  properly  pro- 
portioned, probably  better  than  would  be  gotten  by  any 
other  pump  doing  the  same  service. 

The  industrial  importance  of  this  pump;  the  difficulty 
surrounding  its  theoretic  analysis;  the  diversity  in  practice 
and  results;  the  scarcity  of  literature  on  the  subject;  and  the 
fact  that  no  patent  covers  the  air  lift  in  its  best  form,  seem 
to  justify  the  author  in  giving  it  relatively  more  discussion 
than  is  given  on  some  better  understood  applications  of 
compressed  air. 

Art.  33.     Theory  of  the  Air-lift  Pump. 

An  attempt  at  rational  analysis  of  this  pump  reveals  so 
many  variables,  and  some  of  them  uncontrollable,  that 
there  seems  little  hope  that  a  satisfactory  rational  formula 

49 


50 


COMPRESSED  AIR 


will  ever  be  worked  out.  However,  a  study  of  the  theory 
will  reveal  tendencies  and  better  enable  the  experimenter  to 
interpret  results. 

In  Fig.  11,  P  is  the  water  discharge  or  reduction  pipe  with 
area  a,  open  at  both  ends  and  dipped  into  the  water.  A 
is  the  air  pipe  through  which  air  is  forced  into 
the  pipe,  P,  under  pressure  necessary  to 
overcome  the  head  D.  6  is  a  bubble  liberated 
in  the  water  and  having  a  volume  0  which 
increases  as  the  bubble  approaches  the  top  of 
the  pipe. 

The  motive  force  operating  the  pump  is  the 
buoyancy  of  the  bubble  of  air,  but  its  buoy- 
ancy causes  it  to  slip  through  the  water  with 
a  relative  velocity  u. 

In  one  second  of  time  a  volume  of  water 
=  au  will  have  passed  from  above  the  bubble 
r^jj       to  below  it  and  in  so  doing  must  have  taken 
__\         some  absolute  velocity  s  in  passing  the  con- 

,-,.  tracted  section  around  the  bubble, 

fig.  11. 

Equating  the  work  done  by  the  buoyancy 
of  the  bubble  in  ascending,  to  the  kinetic  energy  given  the 
water  descending  we  have 


s 
wOu  =  wau  —  where  w  =  weight  of  water, 


or 


(a) 


—  is  the  equivalent  of  the  head  h  at  top  of  the  pipe  which 

0 

is  necessary  to  produce  s,  therefore  h  =  -  • 

ci 

Suppose  the  volume  of  air,  0,  to  be  divided  into  an  infinite 
number  of  small  particles  of  air,  then  the  volume  of  a  particle 
divided  by  a  would  be  zero  and  therefore  s  would  be  zero; 
but  the  sum  of  the  volumes  ( =  0)  would  reduce  the  specific 
gravity  of  the  water,  and  to  have  a  balance  of  pressure  be- 
tween the  columns  inside  and  outside  the  pipe  the  equation 


wO  =  wah  must  hold. 


THE  AIR-LIFT  PUMPS  51 

Hence  again  h  =  — ,  showing  that  the  head  h  depends  only 

on  the  volume  of  air  in  the  pipe  and  not  on  the  manner  of 
its  subdivision. 

The  slip,  u,  of  the  air  relative  to  the  water  constitutes 
the  chief  loss  of  energy  in  the  ajr  lift.  To  find  this  apply  the 
law  of  physics,  that  forces  are  proportional  to  the  velocities 
they  can  produce  in  a  'given  mass  in  a  given  time.  The 
force  of  buoyancy  wOf  of  the  bubble  causes  in  one  second  a 
downward  velocity  s  in  a  weight  of  water  wau.  Therefore 

wO       s 


wau       g 

Whence       u  =  —  ®-  .    But  —  =  —  as  proved  above. 
as  a      2  g 


Therefore  tt-«.  (b) 

This  shows  that  the  slip  varies  with  the  square  root  of 
the  volume  of  the  bubble.  It  is  therefore  desirable  to 
reduce  the  size  of  the  bubbles  by  any  means  found  possible. 

If  u  =  ^  ,  then  the  bubble  will  occupy  half  the   cross 
2 

section  of  the  pipe.  This  conclusion  is  modified  by  the 
effect  of  surface  tension,  which  tends  to  contract  the  bubble 
into  a  sphere.  The  law  and  effect  of  this  surface  tension 
cannot  be  formulated  nor  can  the  volume  of  the  bubbles  be 
entirely  controlled.  Unfortunately,  since  the  larger  bubbles 
slip  through  the  water  faster  than  the  small  ones,  they  tend 
to  coalesce;  and  while  the  conclusions  reached  above  may 
approximately  exist  about  the  lower  end  of  an  air  lift,  in 
the  upper  portion,  where  the  air  has  about  regained  its 
free  volume,  no  such  decorous  proceeding  exists,  but  instead 
there  is  a  succession  of  more  or  less  violent  rushes  of  air 
and  foamy  water. 

The  losses  of  energy  in  the  air  lift  are  due  chiefly  to  two 
causes:  first,  the  slip  of  the  bubbles,  through  the  water, 
and  second,  the  friction  of  the  water  on  the  sides  of  the 
pipe.  As  one  of  these  decreases  the  other  increases,  for  by 


52 


COMPRESSED  AIR 


reducing  the  velocity  of  the  water  the  bubble  remains 
longer  in  the  pipe  and  has  more  time  to  slip. 

The  best  proportion  for  an  air  lift  is  that  which  makes  the 
sum  of  these  two  losses  a  minimum.  Only  experiment 
can  determine  what  this  best  proportion  is.  It  will  be 
affected  somewhat  by  the  average  volume  of  the  bubbles. 
As  before  said,  any  means  of  reducing  this  volume  will 
improve  the  results. 

Art.  34.  Design  of  Air-lift  Pumps. 

The  variables  involved  in  proportioning  an  air-lift  pump 


are:  — 


Fig.  12. 


Q  =  volume  of  water  to  be  lifted, 

h  =  effective  lift  from  free    water    surface 

outside  the  discharge  pipe, 
I  =  D  +  h  =  total   length    of    water    pipe 
above  air  inlet, 

D  =  Depth  of  submergence  =  depth  at  which 
air  is  liberated  in  water  pipe  meas- 
ured from  free  water  surface  outside 
the  discharge  pipe. 

va  =  volume  per  second  of  free  air  forced  into 
well, 

a  =  area  of  water  pipe, 

A  =  area  of  air  pipe, 

0  =  volume  of  the  individual  bubbles. 

The  designer  can  control  A,  a,  D  +  h  and 
va  but  he  has  little  control  over  0,  and  cannot 
foretell  what  D  and  Q  will  be  until  the  pump 
is  in  and  tested. 

When  the  pump  is  put  in  operation  the 
free  water  surface  in  the  well  will  always  drop. 
What  this  drop  will  be  depends  first  on  the 


geology  and  second  on  the  amount,  Q,  of  water  taken  out. 
In  very  favorable  conditions,  as  in  cavernous  limestone, 
very  porous  sandstone  or  gravel,  the  drop  may  be  only  a 
few  feet,  but  in  other  cases  it  may  be  so  much  as  to  prove 
the  well  worthless.  In  any  case  it  can  be  determined  by 
noting  the  drop  in  the  air  pressure  when  the  water  begins 


THE  AIR-LIFT  PUMPS  53 

flowing.  If  this  drop  is  p  pounds,  the  drop  of  water  surface 
in  the  well  is  2.3  X  p  feet. 

Unless  other  and  similar  wells  in  the  locality  have  been 
tested,  the  designer  should  not  expect  to  get  the  best  pro- 
portion with  the  first  set  of  piping,  and  an  inefficient  set  of 
piping  should  not  be  left  in  the  \$ell. 

The  following  suggesfions  for  proportioning  air  lifts  have 
proved  safe  in  practice,  but,  of  course,  are  subject  to  revision 
as  further  experimental  data  are  obtained.  (See  Figs.  13 
and  14.) 

Air  Pipe.  Since  in  the  usually  very  limited  space  high 
velocities  must  be  permitted  we  may  allow  a  velocity  of 
about  30  ft.  per  second  in  the  air  pipe. 

Submergence.     The  ratio   •— :•    is  defined  as  the  Sub- 

D  +  h 

mergence  ratio.  Experience  seems  to  indicate  that  this 
should  be  not  less  than  one-half;  and  about  60  per  cent 
is  a  common  rule  of  practice.  Probably  the  efficiency  will 
increase  with  the  submergence.  The  cost  of  the  extra  depth 
of  well  necessary  to  get  this  submergence  is  the  most  serious 
handicap  to  the  air-lift  pump. 

Ratio  -~  • 

Let  D  =  depth  of  submergence  and  h  —  effective  lift  = 
nD.  Then  the  energy  in  the  compressed  air  is 

pava   logeP  +  33-3>),    5-lL33^3 being  the  ratio  of  compres- 
V     33.3     /          33.3 

sion,  =  r,  and  the  effective  work  in  water  lifted  is 

wQh  =  62.5  QnD. 
If  E  be  the  efficiency  of  the  system,  then 

62.5  X  Q  X  nD  =  E  X  2100  va  X  2.3  Iogi0  (r), 
cubic  foot  units  being  used  and  common  logs. 

Whence  ^  =  J_  ^  -JL-.  (27) 

Q       77.3  E  logio  r 

Several  apparently  well  proportioned  wells  are  on  record, 
see  Art.  37,  in  which  D  is  from  350  to  500  feet,  n  about  § 


54 


COMPRESSED  AIR 


and  E  40  to  50  per  cent.     Taking  n 
cent,  Eq.  (27)  reduces  to 


Q       50  logio  r 
From  which  the  following  table  is  computed. 


and  E  =  45  per 
(27a) 


h 

D 

I 

Va 

Q 

h 

D 

I 

Va 

Q 

6.6 
33. 
66. 
100. 
133. 

10 
50 
100 
150 
200 

16.6 
83.3 
166.0 
250.0 
333. 

1.8 
2.5 
3.4 
4.1 

4.8 

167 
200 
233 
267 
300 
333 
366 

250 
300 
350 
400 
450 
500 
550 

417 
500 
583 
667 
750 
833 
916 

5.4 
6.1 
6.6 

7.2 
7.8 
8.4 
8  9 

This  table  is  reproduced  in  the  curve  plate  V.  It  should 
be  used  only  with  full  recognition  of  the  assumptions  on 
which  it  is  based,  and  with  due  regard  to  what  follows 
about  velocities  in  the  water  pipe.  The  table  has  been 
verified  for  h  between  200  and  400  feet.  For  lower  lifts  it 
would  be  expected  that  a  better  efficiency  could  be  obtained— 
the  best  data  that  can  be  found  seem  to  indicate  that  such 
is  the  case.  In  consideration  of  this  the  dotted  line  on 
plate  V  may  be  a  better  guide  than  the  full  line. 

Velocity  in  the  Water  Pipe. 

This  is  the  factor  that  most  affects  the  efficiency,  but  un- 
fortunately, owing  to  the  usual  small  area  in  the  well,  the 
velocity  cannot  always  be  kept  within  the  limits  desired. 
The  complicated  action  and  varying  conditions  in  a  well 
make  the  designer  entirely  dependent  on  the  results  of 
experience  in  fixing  the  allowable  velocities  in  the  discharge 
pipes. 

The  velocity  of  the  ascending  column  of  mixed  air  and 
water  should  certainly  be  not  less  than  twice  the  velocity  at 
which  the  bubble  would  ascend  in  still  water.  This  would 
probably  put  the  most  advantageous  least  velocity  in  any  air 
lift  at  between  five  and  ten  feet,  and  this  would  occur  where 
the  air  enters  the  discharge  pipe. 


THE  AIR-LIFT  PUMPS 


55 


:::" 

4r 

mi 

PLATE  V 
•  -  -  -  -  Curve  representing  T£q.27a 
-  -  The  dotted  line  is  probably 

:::::  g 

1§« 

a 

f-4 

§3 
o 

£ 

—  82 

00                          t-                         CO 

\Q               ft           •*•                             CO                               IN 

"A 

1 
I 

LLLUJo 

COMPRESSED  AIR 
The  velocity  at  any  section  of  the  pipe  will  be 


where  Q  and  v  are  the  volumes  of  water  and  air  respectively 
and  a  the  effective  area  of  the  water  pipe,  s  increases 
from  bottom  to  top  probably  very  nearly  according  to  the 
formula 


(28) 
where 

r  =  ratio  of  compression  under  running  conditions, 
I  =  total  length  of  discharge  pipe  above  air  inlet, 
x  =  distance  down  from  top  of  discharge  pipe  to  section 
where  velocity  is  s. 

The  formula  (28)  is  based  on  the  assumption  that  the  vol- 
ume of  air  varies  as  the  ordinate  to  a  straight  line  while 
ascending  the  pipe  through  length  L  As  the  volume  of  each 
bubble  increases  in  ascending  the  pipe,  the  velocity  of  the 
mixture  of  water  and  air  should  also  increase  in  order  to 
keep  the  sum  of  losses  due  to  slip  of  bubble  and  friction  of 
water  a  minimum;  but  for  deep  wells  with  the  resultant 
great  expansion  of  air  the  velocity  in  the  upper  part  of  the 
pipe  will  be  greater  than  desired,  especially  if  the  discharge 
pipe  be  of  uniform  diameter.  Hence  it  will  be  advantageous 
to  increase  the  diameter  of  the  discharge  pipe  as  it  ascends. 
The  highest  velocity  (at  top)  probably  should  never  exceed 
twenty  feet  per  second  if  good  efficiency  is  the  controlling 
object. 

Good  results  have  been  gotten  in  deep  wells  with  velocities 
about  six  feet  at  air  inlet  and  about  twenty  feet  at  top.  (See 
Art.  37.) 

Fig.  13  shows  the  proportions  and  conditions  in  an  air 
lift  at  Missouri  School  of  Mines. 

The  flaring  inlet  on  the  bottom  should  never  be  omitted. 
Well-informed  students  of  hydraulics  will  see  the  reason  for 
this,  and  the  arguments  will  not  be  given  here. 


THE  AIR-LIFT  PUMPS  57 

The  numerous  small  perforations  in  the  lower  joint  of  the 
air  pipe  liberate  the  bubbles  in  small  subdivisions  and  some 
advantage  is  certainly  gotten  thereby. 

No  simpler  or  cheaper  layout  can  be  designed,  and  it  has 
proved  as  effective  as  any.  It  is  the  author's  opinion  that 
nothing  better  has  been  found  where  submergence  greater 
than  50  per  cent  can  be  had. 

Art.  35.    The  Air  Lift  as  a  Dredge  Pump. 

The  possibilities  in  the  application  of  the  air  lift  as  a 
dredge  pump  do  not  seem  to  have  been  fully  appreciated. 
This  may  be  due  to  its  being  free  from  patents  and  therefore 
no  one  being  financially  interested  in  advocating  its  use. 

With  compressed  air  available  a  very  effective  dredge 
can  be  rigged  up  at  relatively  very  little  cost  and  one  that 
can  be  adapted  to  a  greater  variety  of  conditions  than  those 
in  common  use,  as  the  following  will  show. 

Suggestions: 

Clamp  the  descending  air  pipe  to  the  outside  of  the  dis- 
charge pipe.  Suspend  the  discharge  pipe  from  a  derrick 
and  connect  to  the  air  supply  with  a  flexible  pipe  (or  hose). 
With  such  a  rigging  the  lower  end  of  the  discharge  pipe  can 
be  kept  in  contact  with  the  material  to  be  dredged  by  lower- 
ing from  the  derrick;  the  point  of  operation  can  be  quickly 
changed  within  the  reach  of  the  derrick,  and  the  dredge  can 
operate  in  very  limited  space.  In  dredging  operations  the 
lift  of  the  material  above  the  water  surface  is  usually  small, 
hence  a  good  submergence  would  be  available.  The  depth 
from  which  dredging  could  be  done  is  limited  only  by  the 
weight  of  pipe  that  can  be  handled. 

Art.  36.    Testing  Wells  with  the  Air  Lift. 

The  air  lift  affords  the  most  satisfactory  means  yet  found 
for  testing  wells,  even  if  it  is  not  to  be  permanently  installed. 
Such  a  test  will  reveal,  in  addition  to  the  yield  of  water,  the 
position  of  the  free  water  surface  in  the  well  at  every  stage 
of  the  pumping,  this  being  shown  by  the  gage  pressures. 
However,  some  precautions  are  necessary  in  order  properly 


COMPRESSED   AIR 

to  correct  the  gage  readings  for  friction  loss  in  the  air 
pipe. 

The  length  of  air  pipe  in  the  well  and  any  necessary  cor- 
rections to  gage  readings  must  be  known. 

The  following  order  of  proceeding  is  recommended. 

At  the  start  run  the  compressor  very  slowly  and  note  the 
pressure  pi  at  which  the  gage  comes  to  a  stand.  This  will 
indicate  the  submergence  before  pumping  commences,  since 
there  will  be  practically  no  air  friction  and  no  water  flowing 
at  the  point  where  air  is  discharged.  Now  suddenly  speed 
up  the  compressor  to  its  prescribed  rate  and  again  note  the 
gage  pressure  p2  before  any  discharge  of  water  occurs.  Then 
Pz  —  PI  =  Pf  is  the  pressure  lost  in  friction  in  the  air  pipe. 
When  the  well  is  in  full  flow  the  gage  pressure  p3  indicates 
the  submergence  plus  friction,  or  submergence  pressure  ps  = 
p3  —  p/.  The  water  head  in  feet  may  be  taken  as  2.3  X  p. 
Then,  knowing  the  length  of  air  pipe,  the  distance  down  to 
water  can  be  computed  for  conditions  when  not  pumping 
and  also  while  pumping. 


Art.  37.    Data  on  Operating  Ak  Lifts. 

In  Figs.  13  and  14  are  shown  the  controlling  numerical 
data  of  two  air  -lifts  at  Holla,  Mo.  These  pumps  are  perhaps 
unusual  in  the  combination  of  high  lift  and  good  efficiency. 
The  data  may  assist  in  designing  other  pumps  under  some- 
what similar  circumstances. 

The  figures  down  the  left  side  show  the  depth  from  sur- 
face. The  lower  standing-water  surface  is  maintained 
while  the  pump  is  in  operation;  the  upper  where  it  is  not 
working. 

The  broken  line  on  the  right  shows,  by  its  ordinate,  the 
varying  velocities  of  mixed  air  and  water  as  it  ascends  the 
pipe. 

The  pump  Fig.  13  delivers  120  gallons  per  minute  with  a 


ratio     T^e  air  =  6.0.     The  submergence  is  58  per  cent  and 
Water 

efficiency  =  Net  energy  in  water  lift  =  5Q 
pv  loge  r 


THE   AIR-LIFT   PUMPS 


59 


The  pump  Fig.  14  delivers  290  gallons  per  minute  with 

a   rati0  Free  air  =  5.2.      Submergence  =  64  per   cent  and 
Water 

efficiency  =  Net  energy  in  water  lift  =  45  per  cent 
pv\oger        ' 


coo 


Fig.  13. 


Fig.  14. 


The  volumes  of  air  used  in  the  above  data  are  the  actual 
volumes  delivered  by  the  compressor.  The  volumetric 
efficiencies  of  the  compressors  by  careful  tests  proved  to  be 
about  72  per  cent. 


CHAPTER  VII 

EXAMPLES  AND  EXERCISES 

Art.  38.  The  following  combined  example  includes  a  solu- 
tion of  many  of  the  types  of  problems  that  arise  in  designing 
compressed-air  plants.  The  student  will  find  it  well  worth 
while  to  become  familiar  with  every  step  and  detail  of  the 
solutions,  which  are  given  more  fully  than  would  be  nec- 
essary except  for  a  first  exercise. 

Example  26.  An  air-compressor  plant  is  to  be  installed 
to  operate  a  mine  pump  under  the  following  specifications: 

1.  Volume  of  water  =  1500  gallons  per  minute. 

2.  Net  water  lift  =  430  feet. 

3.  Length  of  water  pipe  =  1280  feet. 

4.  Diameter  of  water  pipe  =10  inches. 

5.  Length  of  air  pipe  =  1160  feet. 

6.  Atmospheric  pressure  =  14.0  pounds  per  sq.  in. 

7.  Atmospheric  temperature  50°  F. 

8.  Loss  in  transmission  through  air  line  =  8  per  cent  of 
the  pv  logg  r  at  compressor. 

9.  Mechanical  efficiency  of  the  pump  =  90  per  cent  as 
revealed  by  the  indicators  on  the  air  end  and  the  known 
work  delivered  to  the  water. 

10.  Average  piston  speed  of  pump  =  200  feet  per  minute. 

11.  Mechanical  efficiency  of  the  air  compressor  =  85  per 
cent  as  revealed  by  the  indicator  cards. 

12.  R.P.M.  of  air  compressor  =  90  and  volumetric  effi- 
ciency =  82  per  cent. 

13.  In  compression  and  expansion  n  =  1.25. 

Preliminary  to  the  study  of  the  problems  involving  the  air 
we  must  determine: 

(a)  Total  pressure  head  against  which  the  pump  must  work. 

By  the  methods  taught  in  hydraulics  the  friction  head  in  a 
pipe  10  inches  in  diameter,  1280  feet  long,  delivering  1500 

60 


EXAMPLES  AND  EXERCISES  61 

gallons  per  minute,  is  about  20  feet.     Therefore  the  total 
head  =  450  feet. 

(b)  Total  work  (W\)  delivered  to  the  water  in  one  minute. 
Wi  =  1500  X  8}  X  450  =  5,625,000  foot-pounds. 

(c)  Total  work  (W)  required  in  air  end  of  pump. 

w 

By  specification  9,  W  =  ^  =  6,250,000    ft.-lbs.  =  190 

.90 

horse  power. 

For  the  purpose  of  comparison,  two  air  plants  will  be 
designed;  the  first,  designated  d,  as  follows: 

(d)  Compression  single-stage  to   80    pounds   gage.     No 
reheating.     No  expansion  in  air  end  of  pump.     Pump  direct 
acting  without  fly  wheels. 

Determine  the  following: 

(dl)   Air  pressure  at  pump  and  pressure  lost  in  air  pipe. 

By  specification  8  and  Eq.  (25), 


14 
Whence,  using  common  logs,  log  ^-  =  0.76118  and 

p2  =  80.78. 

Then  lost  pressure  =  pi  -  p2  =  94  -  80.78  =  13.22  =  f, 
and  gage  pressure  at  pump  =  80  -  13.22  =  66.78. 

(d2)    Ratio  between  areas  of  air  and  water  cylinders  in  pump. 

The  pressure  due  to  450  feet  head  =  450  X  .434  =  194.3, 
say  195  pounds,  per  sq.  in.;  and  since  pressure  by  area  must 

be  equal  on  the  two  ends,     area  air  en^-  =  -^-  =3  nearly. 

area  water  end      66.78 

(d3)  Volume  of  compressed  air  used  in  the  pump.  Cubic 
feet  per  minute: 

Evidently  from  solution  (d2)  the  volume  of  compressed 
air  used  in  the  pump  will  be  three  times  that  of  the  water 
pumped,  or 

ISO?  x3  =  601.6  cu.  ft.  per  min. 


62  COMPRESSED  AIR 


Diameters  of  air  cylinder  and  of  water  cylinder. 
Since  the  piston  speed  is  limited  to  200  feet  per  min. 
(spec.  10)  and  the  volume  is  1500  gallons,  we  have,  when  all  is 
reduced  to  inch  units  and  letting  a  =  area  of  water  cylinder, 
a  X  200  X  12  =  1500  X  231.  Whence  a  =  144  sq.  in.  which 
requires  a  diameter  of  about  13f  inches. 

The  area  of  air  cylinder  is  by  d%  three  times  that  of  the 
water  cylinder,  which  gives  a  diameter  23  J  inches  for  the  air 
cylinder. 

(c£5)    Volume  of  free  air. 

From  dl,  r  at  the  pump  =  5.76.     Therefore 

va  =  601.6  X  5.76  =  3465  cu.  ft.  per  min. 
Diameter  of  air  pipe. 


The  mean  r  in  the  air  pipe  is  5'76  +  6'72  =  6.24.     Using 

2i 

this  in  Eq.  (21)  with  c  =  .06,  we  get  d  =  5  inches. 

Or  using  plate  III  with  r  X  13.22  -r-  1.160   or  r  X  ^^ 

JL  •  XOvJ 

as  vertical  ordinate  and  3465  as  horizontal  ordinate,  the  in- 
tersection falls  near  the  5-inch  line. 

(dl)    Horse  power  required  in  steam  end  of  compressor. 

By  table  II  the  weight  per  foot  of  free  air  is  .07422  pound 
per  cu.  ft.  Total  weight  of  air  compressed  =  Q 

Q  =  .07422  X  3465  =  257  pounds  per  min. 

In  table  I  opposite  r  =  6.72  in  column  9  find  by  inter- 
polation .3736.  Then 

Horse  power  =  2.57  X  .3736  X  (460  +  50)  =  489.6  in  air 

end  =  —  -^-  =  576  in  steam  end. 

.85 

The  second  plant  will  be  designated  by  the  letter  e  and 
will  be  two-stage  compression  to  200  pounds  gage  at  air 
compressor,  will  be  reheated  to  300°  at  the  pump  and  used 
expansively  in  the  pump;  the  expansion  to  be  such  that  the 
temperature  will  be  32°  at  end  of  stroke. 

(el)   Air  pressure  at  pump. 

Apply  Eq.  (25)  as  in  dl.  In  this  case  n  (at  the  com- 
pressor) =  15.3  and  r2  (at  the  pump)  =  12.3.  Therefore 


EXAMPLES  AND  EXERCISES  63 

pressure   at  the  pump  =  12.3  X  14  =  172.3   and  the  lost 
pressure  =  214  -  172.3  =  41.7-  =  f. 

(e2)    Point  of  cut-off  in  air  end  of  pump  =  fraction  of 
stroke  during  which  air  is  .admitted. 

t       /n\n~1 
By  Eq.  (11)  viz.  -  =  (-1)      v  ,  putting  in  numbers  we  get 


12?  =  f&V*  whence  -1  =  .176,  which  is  the  point  of  cut-off, 
760      \v2/  v2 

and  v2  =  5.68  vi. 

760 

Or  go  into  table  I  in  column  5,  find  the  ratio  —  —  -  =   1.545, 

49.2 

and  in  same  horizontal  line  in  column  3  find  .176. 

(e3)  Volume  of  compressed  hot  air  admitted  to  air  end  of 
pump. 

Apply  Eq.  (9)  viz.  Work  =  P&^-M*  +  ^  _  ^ 

n  —  I 

In  this  we  have  Work  =  6,250,000,  v2  =  5.68  v1}  pi  =  214, 
n  —  1  =  .25,  pa  =  14,  and  p2  must  be  found  by  Eq.  (lla),  or 
it  may  be  gotten  from  table  I  by  noting  that  for  a  tempera- 

ture ratio  of  1.545  the  pressure  ratio  is  8.8  and  -  =  .1136, 

r 

therefore  pz  =  .1136  X  172.3  =  19.57.     This  would  give  gage 
pressure  =  5.57. 

Inserting  these  numerals  in  Eq.  (9)  we  get 

6,250,000  =  144  Vl  /ll2-3"  5-68X19.57  +172.3_14X5.68Y 
\  -25  / 

Whence  Vi  =  128.6  cu.  ft.  per  min. 

(e4)  Diameter  of  air  cylinder  of  pump  when  air  and  water 
pistons  are  direct  connected. 

Since  expansion  ratio  is  5.68  (see  e2)  and  the  volume  before 
cut-off  is  128.6,  the  total  piston  displacement  is  128.6  X  5.68  = 
730.8  cu.  ft.  per  min.  When  the  air  and  water  pistons  are 
direct  connected  they  must  travel  through  equal  distances, 
therefore  the  air  piston  travels  through  200  ft.  per  min.  (spec. 
10).  Then  if  a  =  area  of  piston  in  sq.  ft.  we  have 

200  a  =  730.8     and     a  =  3.654  sq.  ft. 
By  table  X  the  diameter  is  26  inches  nearly. 


64  COMPRESSED   AIR 

(e5)  Volume  of  cool  compressed  air  used  by  pump,  cu.  ft. 
per  min. 

By  e3  the  volume  of  hot  compressed  air  is  128.6,  and  since 
under  constant  pressure  volumes  are  proportional  to  abso- 
lute temperatures,  we  have 

ci  n 

=  ^~  -  Whence  v  =  86.3  cu.  ft.  per  min. 


128.6       760 

(e6)    Volume  of  free  air  used. 

From  el  the  ratio  of  compression  at  the  pump  is  12.3  and 
from  e5  the  volume  of  cool  compressed  air  is  86.3,  therefore 
the  volume  of  free  air  is  86.3  X  12.3  =  1061.6. 

(el)    Diameter  of  air  pipe. 

The  r  for  Eq.  (21)  is  12-3  +  15-3   =  13.8. 

Applying  Eq.  (21)  with  coefficient  c  =  .07  we  have 
/  07  X  1160 


(eS)    Horse  power  required  in  steam  end  of  compressor. 

By  dl  the  weight  per  cu.  ft.  of  free  air  is  .07422  and  by  eQ 
the  volume  of  free  air  compressed  is  1061.6.  Therefore  the 
total  weight  compressed  is  .07422  X  1061.6  =  78.8  pounds 
per  min.,  and  the  initial  absolute  temperature  is  510. 

In  the  two-stage  compression  r2  =  15.3,  and  assuming  equal 
work  in  the  two  stages  the  rx  =  Vl5.3  =  3.91  nearly. 
(See  Art.  12.)  Then  going  into  Table  I  with  r  =  3.91  in  column 
9  find  .2525.  Hence  horse  power  =  .2525  X  78.8  X  510  = 
101.5  for  one  stage,  and  for  the  two  stages  101.5  X  2  =  203, 

203 
and  (spec.  11)  -   =  238.8  horse  power  in  steam  end. 

.85 

(e9)  Diameter  of  air  compressor  cylinders,  assuming  3- 
foot  strokes  and  2%-inch  piston  rods,  equal  work  in  the  two 
cylinders  and  allowing  for  volumetric  efficiency. 

By  eft  the  free  air  volume  is  1061.6  and  (spec.  12)  the 
volumetric  efficiency  =  82  per  cent.  Therefore  the  piston 

displacement  =  -  ^—L-  =  1294.6  cu.  ft.  per  min. 

.82 


EXAMPLES  AND  EXERCISES  65 

By  spec.  12  the  R.P.M.  =  90.  Therefore  the  displace- 
ment per  revolution  =  14.7,  nearly,  for  the  low-pressure 
cylinder.  Add  to  this  the  volume  of  one  piston  rod  length 
of  3  feet,  which  is  3  X  .0341  =  0.1023.  Whence  the  volume 
per  revolution  must  be  14.8  or  for  one  stroke  7.4.  Whence 

the  area  =  ^-  =  2.466  sq.  ft.  <J3y  Table  X  the  diameter 

o  » 

is  21  J  inches  nearly  for  low-pressure  cylinders. 

The  high-pressure  cylinder  must  take  in  the  net  volume 
of  air  compressed  to  r  =  3.91  (see  e8).  Therefore  the  net 


volume  per  revolution  =  —  -  ^—  =  3.02.    Add  one  piston 

90  X  o.91 

rod  volume  and  get  3.12  per  revolution  or  1.56  per  stroke 
and  an  area  of  0.53  sq.  ft.  By  Table  X  this  requires  a 
diameter  of  10  inches  nearly. 

(elO)    Temperature  of  air  at  end  of  each  compression  stroke. 

In  Table  I  the  ratio  of  temperatures  for  r  =  3.91  is  1.313. 
Hence  the  higher  temperature  =  510  X  1.313  =  669  absolute 
**=  209  F. 

EXERCISES 

i  .  (a)  Assuming  isothermal  conditions,  how  many  revo- 
lutions of  a  compressor  16"  stroke,  14"  diameter,  double 
acting,  would  bring  the  pressure  up  to  100  Ibs.  gage  in  a 
tank  4  feet  diam.  X  12  feet  length,  atmospheric  pres- 
sure =  14.5  per  sq.  in.? 

(b)  What  would  be  the  horse  power  of  such  a  compressor 
running  at  100  R.P.M.  ? 

(c)  What  would  be  the  horse  power  if  the  compression 
were  adiabatic  ? 

(d)  What  weight  of  air  would  be  passed  per  minute  when 
R.P.M.  =  100  and  T  =  60°  F.  ? 

2.  The  air  end  of  a  pump  (operated  by  compressed  air)  is 
20"  diam.  by  30"  stroke,  R.P.M.  =  50,  cut-off  at  \  stroke, 
free  air  pressure  =  14.0,  Ta  =  60°,  compressed  air  delivered 
at  75  Ibs.  gage,  T  =  60°  and  n  =  1.41. 

(a)  Find  work  done  in  horse  power. 

(b)  Find  weight  handled  per  minute. 

(c)  Find  temperature  of  exhaust  (degrees  F). 


66  COMPRESSED  AIR 

3.  With  atmospheric  pressure,  pa  =  14.7,  and  Ta  =  50°, 
under  perfect  adiabatic  compression,  what  would  be  the  pres- 
sure (gage)  and  temperature  (F.)  when  air  is  compressed  to 

(a)  |  its  original  volume  ? 

(6)  i  its  original  volume  ? 

(c)  £  its  original  volume  ? 

(d)  I  its  original  volume  ? 

(e)  TU  its  original  volume  ? 

4.  With  pa  =  14.1  and  Ta  =  60°  what  will  be  the  pres- 
sure of  a  pound  of  air  when  its  volume  =  3  cu.  ft.  ? 

5.  What  would  be  the  theoretic  horse  power  to  compress 
10  pounds  of  air  per  minute  from  pa  =  14.3  and  Ta  =  60°  to 
90  pounds  gage? 

(a)    Compression  isothermal. 
(6)    Compression  adiabatic. 

6.  Find  the  point  of  cut-off  when  air  is  admitted  to  a  motor 
at  250°  F.  and  expanded  adiabatically  until  the  temperature 
falls  to  32°  F. 

7.  What  is  the  weight  of  1  cu.  ft.  of  air  when  pa  =  14.0 
and  Ta  =  -  10°  ? 

8.  A  compressor  cylinder  is  20"  diam.  by  26"  stroke  double 
acting.     Clearance  =  0.8%,  piston  rod  =  2",  R.P.M.  =  100, 
atmospheric  pressure,  pa  =  14.3,  atmospheric  temperature 
=  Ta  =  60°  F.,  and  gage  pressure  =  98  Ibs. 

Determine  the  following: 
(a)    Compression  isothermal. 

la.    Volume  of  free  air  compressed,  cu.  ft.  per  min. 

2a.    Volume  of  compressed  air,  cu.  ft.  per  min. 

3a.   Work  of  compression,  ft.-lbs.  per  min. 

4a.    Lbs.  of  cooling  water,  7\  =  50°,  T2  =  75°. 
(6)    n  =  1.25  and  air  heated  to  100°  while  entering. 

16.   Volume  of  free  air  compressed  per  min. 

26.   Volume  of  cool  compressed  air  per  min. 

36.   Work  done  in  compression. 

46.    Temperature  of  air  at  discharge. 

9.  The  cylinder  of  a  compressed-air  motor  is  18"  X  24", 
the  R.P.M.  =  90,  air  pressure   100  pounds  gage.     In  the 


EXAMPLES  AND  EXERCISES 


67 


motor  the  air  is  expanded  to  four  times  its  original  volume 
(cut-off  at  J),  with  n  =  1.25. 

(a).  Determine  the  horse  power  and  final  temperature 
when  initial  T  =  60°  F. 

(6) .  Determine  the  horse  power  and  final  temperature  when 
initial  T  =  212°  F. 

10.  Observations  on~  an  air  compressor  show  the  intake 
temperature  to  be  60°  F.,  the  r  =  7  and  the  discharge  tem- 
perature =  300  F.     What  is  the  n  during  compression  ? 

Hint.    Use  Eq.  (lla)  with  n  unknown. 

11.  In  a  compressed-air  motor  what  percentage  of  power 
will  be  gained  by  heating  the  air  before  admission  from 
60°  to  300°  F.  ? 

12.  If  air  is  delivered  into  a  motor  at  60°  F.  and  the  ex- 
haust temperature  is  not  to  fall  below  32°  F.,  what  ratio  of 
expansion  can  be  allowed  ?     What  could  be  allowed  if  initial 
temperature  were  300°  ?     What  would  be  the  ratio  of  work 
gotten  in  the  two  cases  assuming  n  =  1.25  ? 

13.  A  compressed-air  locomotive  system  is  estimated  to 
require  4000  cu.  ft.  per  min.  of  free  air  compressed  to  500 
pounds  gage  in  three  stages  with  complete  cooling  between 
stages. 

Assume  n  =  1.25,  pa  =  14.5,  Ta  =  60°,  Vol.  Eff.  =  80  per 
cent,  Mechanical  Eff.  =  85  per  cent  and  R.P.M.  =  60. 

Compute  the  volume  of  piston  stroke  in  each  of  the  three 
cylinders  and  the  total  horse  power  required  of  the  steam  end. 

14.  A  compressor  is  guaranteed  to  deliver  4  cu.  ft.  of  free 
air  per  revolution  at  a  pressure  of  116  (absolute).     To  test 
this  the  compressor  is  caused  to  deliver  into  a  closed  system 
consisting  of  a  receiver,  a  pipe  line  and  a  tank.     Observed 
conditions  are  as  follows:    . 


Receiver. 

Pipe. 

Tank. 

Pressures  at  start  (ab  ) 

14  5 

14  5 

14  5 

Temperatures  at  start  (F.)  
Pressures  at  end  (ab.)  
Temperatures  at  end  (F  ) 

60.0 
116.0 
150  0 

60.0 
116.0 
90  0 

60.0 
116.0 
60  0 

Volumes  (cu.  ft.) 

50  0 

10  0 

100  0 

68  COMPRESSED   AIR 

How  many  revolutions  of  the  compressor  should  produce 
this  effect  ? 

15.  Find  the  discharge  in  pounds  per  minute  through  a 
standard  orifice  when  d  =  2",  i  —  5",  t  =  600°  and  pa  = 
14.0. 

16.  What  diameter  of  orifice  should  be  supplied  to  test 
the  delivery  of  a  compressor  that  is  guaranteed  to  deliver 
1000  cu.  ft.  per  min.  of  free  air  ? 

17.  What  is  the  efficiency  of  transmission  when  air  pres- 
sure drops  from  100  to  90  pounds  (gage)  in  passing  through 
a  pipe  system  ? 

18.  A  compressor  must  deliver  100  cu.  ft.  per  min.  of  com- 
pressed air  at  a  pressure  =  90  pounds,  gage,  at  the  terminus 
of  a  pipe  3000  ft.  long  and  3"  diameter.     pa=  14.4,  Ta  = 
60°  F. 

(a)  Assuming  a  Vol.  Eff.  =  75  per  cent,  what  must  be  the 
piston  displacement  of  the  compressor  ? 

(6)    What  pressure  is  lost  in  transmission  ? 

(c)  What  horse  power  is  necessary  in  steam  end  of  com- 
pressor if  n  =  1.25  and  the  mechanical  efficiency  =  85  per 
cent? 

(d)  What  would  be  the  efficiency  of  the  whole  system 
if  air    is    applied    in    the    motor    without   expansion,    the 
efficiency  to  be  reckoned  from  steam  engine  to  work  done 
in  motor  ? 

19.  It  is  proposed  to  convey  compressed  air  into  a  mine 
a  distance  of  5000'.     The  question  arises:  Which  is  better, 
a  3"  or  a  4"  pipe? 

Compare  the  propositions  financially,  using  the  following 
data:  Nominal  capacity  of  the  plant  =  1000  cu.  ft.  free 
air  per  min.,  Vol.  Eff.  of  compressor  =  80  per  cent,  n  =  1.25 
gage  pressure  at  compressor  =  100,  weight  of  free,  air  wa  = 
.074,  pa  =  14.36,  weight  of  3"  pipe  =  7.5  and  of  4"  pipe  = 
10.7  pounds  per  foot.  Cost  of  pipe  in  place  =  4  cents  per 
pound.  Cost  of  one  horse  power  in  form  of  pv  log  r  for  10 
hours  per  day  for  one  year  =  $150.  Plant  runs  24  hours  per 
day.  Rate  of  interest  =  6  per  cent. 


EXAMPLES  AND  EXERCISES  69 

20.  Air  enters  a  4"  pipe  with  60  feet  velocity  and  80 
pounds  gage  pressure;  the  air  pipe  is  1500  feet  long;  pa  =  14.7. 

(a)  Find  the  efficiency  of  transmission. 

(b)  Find  horse  power  delivered  at  end  of  pipe  in  form 
pv  log  r  when  T  =  60°  F. 

(c)  Find  horse  power  delivered  at  end  of  pipe  in  form 
P0Xv. 

21.  An  air  pipe  is  to  be  2000  feet  long  and  must  deliver  50 
horse  power  at  the  end  with  a  loss  of  5  per  cent  of  the  pv  log  r 
as  measured  at  compressor.     The  pressure  at  compressor  is 
75  pounds  gage.     pa  =  14.7.     Find  diameter  of  pipe. 

22.  Modify  21  to  read:  50  horse  power  .  .  .  with  loss  of 
5  per  cent  of  the  energy  in  form  Pg  X  v,  where  Pg  is  gage 
pressure,  and  find  diameter  of  air  pipe. 

23.  In  case  21  let  pressure  at  compressor  be  250  pounds 
gage  and  find  diameter  of  air  pipe. 

24.  The  air  cylinder  of   a  compressed-air  pump  is  20" 
diam.  by  30"  stroke.     The  machine  is  double  acting  and 
makes  50  R.P.M.     The  cut-oft0  is  to  be  so  adjusted  that  the 
temperature  of  exhaust  shall  be  30°.     pa  =  14.5  and  the  r  at 
pump  =  8. 

(a)  Find  cut-off  when  initial  temperature  is  60°  F. 

(b)  Find  cut-off  when  initial  temperature  is  250°  F. 

(c)  Find  horse  power  in  case  (a) . 

(d)  Find  horse  power  in  case  (b). 

(e)  In  case  (a)  find  efficiency  in  applying  the  pv  log  r  of 
cool  air. 

(/)     In  case  (b)  find  efficiency  in  applying  the  pv  log  r  of 
cool  air. 

(g)    Find  the  volumes  of  free  air  used  in  cases  (a)  and  (b). 

25.  A  compound  mine  pump  is  to  receive  air  at  150  Ibs. 
gage;  this  is  to  be  reheated  from  60°  to  250°  F.,  let  into  the 
H.P.  cylinder  of  the  pump  and  expanded  until  the  temperature 
is  32°,  then  exhausted  into  an  interheater  where  the  tempera- 
ture is  again  brought  to  250°.     It  then  goes  into  the  L.P. 
cylinder  and  is   expanded    down   to   atmospheric   pressure 
=  14.5,  (ab.). 

(a)    Find  point  of  cut-off  in  each  cylinder,  n  =  1.25. 


70  COMPRESSED  AIR 

(6)  If  the  air  is  compressed  in  two  stages  with  n  =  1.25, 
what  will  be  the  efficiency  of  the  system,  neglecting  friction 
losses  \  and  (. 

(c)  How  much  free  air  will  be  required  to  operate  the 
pump  if  it  is  to  deliver  250  horse  power,  assuming  the  efficiency 
of  the  pump  to  be  80  per  cent  reckoned  from  the  work  in  the 
air  end  ? 

(d)  If  the  pump  strokes  be  60  per  min.  and  60"  long,  fix 
diameters  of  air  cylinders  in  case  (c). 

26.  Compute  the  horse  power  of   a   motor   passing   one 
pound  of  air  per  minute  admitted  at  200°  F.   and   116.0 
pounds  (ab.)  r  =  8,  the  air  to  be  expanded  until  pressure 
drops  to  29  pounds  (ab.),  r  =  2. 

27.  A  pump  to  be  operated  by  compressed  air  must  deliver 
1000  gallons  of  water  per  minute  against  a  net  head  of  200' 
through  800'  of  10"  pipe.    The  pump  is  double  acting,  30" 
stroke,  50  strokes  per  min.     The  air  is  reheated  to  275°  F. 
before  entering  the  pump.     The  cut-off  is  so  adjusted  that 
with  n  =  1.25  the  temperature  at  exhaust  =  36°  F.    Mec.  Effi. 
of  pump  =  80%.     Air  pressure  at  compressor  =  90  pounds 
gage,  pa  =  14.4.     Length  of  air  pipe  =  2000'.     Permissible 
loss  in  transmission  =  7  per  cent  of  the  pv  log  r  at  com- 
pressor.    Mec.  Effi.  of  compressor  =  85  per  cent.     Vol.  Effi. 
=  80  per  cent. 

(a)    Proportion  the  cylinders  of  the  pump. 
(6)    Determine  the  volume  of  free  air  used. 
(c)    Determine  the  diameter  of  air  pipe. 

28.  Compare  the  volume  displacement  of  two  air  com- 
pressors, one  at  sea  level  and  the  other  at  12,000  feet  eleva- 
tion; the  compressors  to  handle  the  same  weight  of  air. 

29.  (a)    An  exhaust  pump  has  an  effective  displacement  of 
3  cu.  ft.  per  revolution.     How  many  revolutions  will  reduce 
the  pressure  in  a  gas  tank  from  30  to  5  pounds  absolute? 
Volume  of  tank  =  400  cu.  ft.  when  pa  =  14.7  ? 

(6)  If  the  pump  is  delivering  the  gas  under  a  constant 
pressure  of  30  pounds,  what  is  the  maximum  rate  of  work 
done  by  the  pump  —  foot  pounds  per  revolution? 


ft 

4 


PLATES  AND  TABLES 


NOTES   ON  TABLE  I. 

The  table  is  the  solution  of  formulas  n,  na,  8a  and  la. 

When  the  weight  of  air  passed  and  its  initial  temperature  are  known, 
the  table  covers  all  conditions  including  elevation  above  sea  level,  reheating, 
and  compounding. 

In  compounding,  either  compression  or  expansion,  the  same  weight 
goes  through  each  cylinder.  Then  knowing  the  initial  /  and  the  r  for  each 
cylinder,  find  from  the  table  the  work  done  in  each  cylinder  and  add.  Usu- 
ally the  r  and  t  are  assumed  the  same  for  each  cylinder  —  then  take  out  the 
work  for  one  stage  and  multiply  by  the  number  of  stages. 

The  table  does  not  include  friction  in  the  machine  nor  the  effect  of  clear- 
ance in  expansion  motors. 

The  table  is  equally  applicable  to  compression  or  expansion  provided 
the  correct  r  be  taken  in  cases  of  expansion. 

Example.  Air  is  received  at  such  a  pressure  that  r  =  8.  What  should  be 
the  cut-off  in  order  that  the  temperature  drop  from  60°  to  32°  F.  ?  Expan- 
sion adiabatic. 

The  ratio  of  temperatures  is  1.057,  which  by  linear  interpolation  corre- 
sponds to  a  volume  ratio  of  .871  or  cut-off  at  about  f-. 

What  would  be  the  pressure  at  exhaust? 

The  two  ratios  above  correspond  to  a  —  =  .825.      Therefore  the  final 

pressure  is  .825  X  initial  pressure. 

To  find  the  foot-pounds  of  work  per  pound  of  air  compressed  multiply 
the  number  opposite  the  r  in  column  7,  8  or  n  as  the  case  may  be  by  the 
absolute  initial  temperature,  /. 

To  find  the  weight  compressed,  go  into  Table  II  with  known  atmos- 
pheric conditions  and  the  cubic  feet  capacity  of  the  machine. 

To  find  the  horse  power  per  hundred  pounds  of  air  passed  per  minute, 
multiply  the  number  opposite  r  in  column  9, 10  or  12  as  the  case  may  be  by 
the  absolute  lower  temperature,  t. 


TABLE  I.     GENERAL    TABLE    RELATING    TO    AIR    COM- 
PRESSION   AND    EXPANSION 


Ratio  of 

Work  Factor. 

for  Isothermal 

§ 

Ratio  of 

Greater  to 

Air  Heated  by  Compression. 

Compression. 

1 

g  "3 

Less  to 

Less  Tem- 

o 

g  § 

S     0 

o  0 

Greater 

perature.  — 

n 

HP     Par- 

0 

II 

Volume  — 

Tempera- 

V33.X7~J 

.  r  .  1;  ac~ 

tor  per  100 

u   ^ 

w.    C 

ai 

cS  1 

£  W 

Is 

in     \ 

CO 

JS  I 

Temp. 

Changing. 
.1 

solute. 

H^.j 

Pounds  per 
Minute 
K 

iis  S1 

U 

ta 

O  j3 

vi  =  \¥j 

**  =(r)~T~ 

Factor  K  for  one 

33° 

IT3  § 

cu  ^ 

1 

'1 

pound. 

i 

E 

PH 

n  = 

n  = 

n  = 

n  = 

n  = 

n  = 

K 

1-25 

1.  41 

i-25 

1.41 

n=  1.25 

w-i.4i 

1-25 

1.41 

33° 

r 

I 

V2 

h 

h 

Ft.-Lbs. 

Ft.-Lbs. 

H.P. 

H.P. 

Ft.-Lbs. 

H.P. 

r 

v\ 

VL 

i\ 

t\ 

I  .  OOOO 

I.OOO 

I.OOO 

I.OOO 

I.OOO 

0.0 

0.0 

0 

0 

0.0 

.0 

.  i 

.9091 

.927 

•935 

1.019 

1.028 

5-I3t 

5-140 

•0155 

0155 

5.068 

•0153 

.  2 

.8333 

.862 

.877 

1.037 

1.054 

9.863 

9-932 

.0298 

0301 

9.694 

.0293 

•  3 

.7692 

.812 

.830 

1.054 

1.079 

14.329 

14.450 

•0434 

•0437 

I3.950 

.0422 

•  4 

•7143 

.764 

.787 

1.070 

1.103 

18.503 

18.766 

.0560 

.0568 

17.890 

.0542 

•5 

.6667 

•  723 

•750 

1.085 

1.125 

22.465 

22.827 

.0680 

.0691 

21-559 

.0653 

.6 

•  6250 

.687 

.717 

I.  100 

1.146 

26.186 

26.704 

•0793 

.0809 

24.991 

•°757 

•  7 

•  5882 

.654 

.686 

.112 

1.166 

29-775 

30.417 

.0902 

.0921 

28.214 

•0855 

.8 

•5555 

.625 

•659 

.125 

1.186 

33.178 

33-985 

.1005 

.  1029 

31-252 

.0947 

1.9 

•5263 

.598 

•634 

•137 

1.205 

36.421 

37.422 

.1104 

.1134 

34-127 

•1034 

2.0 

.5000 

•574 

.612 

.149 

1.223 

39.530 

40.733 

.  1198 

•1235 

36-855 

.1117 

2.  I 

.4762 

.552 

•590 

.160 

1.240 

43-897 

.1289 

.133° 

39.450 

.  1196 

2.2 

•4545 

•532 

•571 

.171 

1-259 

45-407 

46.988 

•1376 

.1424 

41.912 

.1270 

2-3 

.4348 

•553 

.  181 

1.273 

48.199 

49.970 

.1461 

.1514 

44-287 

•1342 

2.4 

.4166 

'496 

•537 

.191 

1.289 

-50.884 

52-878 

.1542 

.1602 

46.548 

.1411 

2-5 

.4000 

.480 

•  522 

.202 

1.304 

53-462 

55.676 

.  1620 

.1687 

48.720 

.1476 

2.6 

.3846 

.466 

•  508 

.211 

1-319 

55.988 

58.402 

.1697 

.1769 

50-805 

•1539 

2-7 

•37°4 

•452 

•493 

.220 

1-334 

58.434 

61.054 

.1771 

.1850 

52.811 

.1600 

2.8 

•3571 

•439 

.481 

.229 

1-348 

60.800 

63-651 

•1843 

.1929 

54-745 

.1659 

2.9 

.3448 

•  427 

.469 

•237 

1.362 

63.086 

66.175 

.1912 

.2006 

56.612 

.1715 

3-° 

•3333 

•415 

.458 

.246 

1-375 

65-319 

68.626 

•  1979 

.2080 

58.414 

.1770 

3-1 

.3226 

.405 

.448 

.254 

1.388 

67.499 

71.158 

.2045 

.2156 

60.157 

.1823 

3-2 

•3I25 

•394 

.438 

.262 

1.401 

69.626 

73-400 

.2110 

.2224 

61.845 

.1874 

3-3 

•  303° 

.385 

.428 

.270 

1.414 

71.700 

75-686 

•2173 

.229^ 

63-481 

.1924 

3-4 

.2941 

•376 

.419 

•277 

1.426 

73.720 

77-936 

.2234 

.2362 

65.087 

•1972 

3-5 

•2857 

•367 

.411 

.285 

1.438 

75-688 

80.131 

.2294 

.2428 

66.610 

.2019 

3-6 

•2778 

•359 

•403 

.292 

1.45° 

77.628 

82.307 

•2352 

.2494 

68.108 

.2064 

3-7 

.2703 

•351 

•395 

•299 

1.461 

79-5l6 

84.411 

.2410 

•2557 

69.564 

.2108 

3-8 

.2632 

•343 

•388 

.306 

1-473 

81.350 

86.496 

•2465 

.2621 

70.982 

.2151 

3-9 

.2564 

•337 

.381 

•313 

1.484 

83-158 

88.544 

.2520 

.2683 

72.364 

•2193 

4.0 

.2500 

.33° 

•374 

.319 

1-495 

84.939 

90.510 

•2574 

•2743 

73.710 

•2234 

4.1 

.2439 

.323 

•367 

.326 

1.506 

86.694 

92.472 

.2627 

.2802 

75.023 

.2274 

4.2 

.2381 

•361 

L332 

1.516 

88.395 

94-434 

.2678 

.2862 

76.304 

•2312 

4-3 

.2326 

.311 

•355 

1-339 

1.526 

90-043 

96-346 

.2729 

.2919 

77-555 

•2350 

4.4 

.2273 

.306 

•349 

•345 

1-537 

91.691 

98.202 

.2779 

.2976 

78.776 

.2387 

4-5 

.2222 

.300 

•344 

•35i 

1-547 

93.312 

100.012 

.2828 

•3031 

79.972 

.2424 

4.6 

.2174 

•295 

.338 

•357 

1-557 

94.882 

101.823 

.2875 

•3085 

81.141 

•2459 

4-7 

.2128 

.290 

•333 

•363 

1.566 

96.424 

103.616 

.2922 

.3140 

82  .  284 

.2494 

4.8 

.2083 

.285 

.328 

1.368 

1.576 

97.966 

105.371 

.2969 

•3193 

83.404 

.2528 

73 


TABLE   I    (Continued}. 


I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

12 

4-9 

.204 

.280 

•  324 

i-374 

1-58 

99.481 

107.109 

.301 

.3246 

84.500 

.2561 

S-° 

.2000 

.27 

•3i9 

i.38c 

1-59 

100.943 

108.811 

•305< 

•3297 

85.574 

.2593 

5- 

.196 

.27 

•3i5 

1.385 

i.  60 

102  .  405 

110.493 

•3103 

•3348 

86.627 

.2625 

5- 

.1923 

.26 

.310 

i-39i 

1.61 

103.841 

112.157 

.3147 

•339s 

87.660 

.2657 

5-3 

.188 

.263 

.306 

1.396 

1.62 

105.260 

113.830 

.3180 

-3449 

88.673 

.2687 

5-4 

.185 

.259 

.302 

1.401 

1.63 

106.673 

115.440 

•3232 

.3498 

89.666 

•2717 

5- 

.l8l8 

.256 

.298 

1.406 

1.64 

IO8.OI3 

117.010 

.3273 

-3546 

90.642 

.2747 

5- 

.1786 

.252 

.294 

1.411 

1.64 

109.353 

118.570 

•3314 

•3593 

91.600 

.2776 

5- 

•1754 

.248 

.291 

1.416 

1.65 

110.683 

120.114 

-3354 

.3640 

92.541 

.2805 

S-8 

.1722 

.245 

.287 

1.421 

1.66 

II2.O03 

121.632 

•3394 

.3686 

93.466 

.2833 

5-9 

•1695 

.242 

.284 

1.426 

1.67 

"3  -3°5 

123.150 

•3433 

•3732 

94-375 

.2860 

6.0 

.166 

.238 

.280 

i.43i 

1.68 

114.581 

124.640 

•3472 

•3777 

95-27I 

.2887 

6. 

.1639 

•235 

.277 

1.436 

1.689 

115-831 

126.113 

•3510 

.3822 

96.147 

.2914 

6.2 

.l6l3 

.232 

.274 

1.440 

1.69 

117.080 

127.576 

-3548 

.3866 

97.012 

.2940 

6-3 

.1587 

.229 

.271 

1-445 

i-7°5 

118.303 

129.030 

-3585 

.3910 

97.863 

.2966 

6.4 

.1562 

.226 

.268 

1.449 

1-713 

"9-573 

130.466 

.3622 

•3953 

98.700 

.2991 

6.5 

.1538 

.22; 

.265 

1.454 

1.72 

120.723 

131.880 

-3658 

•3997 

99-524 

.3016 

6.6 

•1515 

.221 

.262 

1.458 

1.728 

121.920 

i33-3oo 

-3694 

•4039 

100.336 

.3040 

6.7 

.1492 

.219 

•259 

1.464 

1.736 

123.063 

134.710 

•3729 

.4082 

101.134 

•3o65 

6.8 

.1471 

.2l6 

.256 

1.467 

1.744 

124.205 

136.090 

.3764 

.4124 

101.920 

.3088 

6.9 

.1449 

.21' 

•  254 

1.471 

i-75 

125-348 

137-450 

•3799 

.4165 

102.700 

.3112 

7.0 

.1428 

.211 

.251 

1.476 

1-758 

126.492 

138.800 

.3833 

.4206 

103.465 

.3135 

7-i 

.  1408 

.2O8 

.249 

1.480 

1.766 

127.608 

140.120 

•3867 

.4246 

104.219 

•3I58 

7.2 

.I389 

.206 

.246 

1.484 

1-773 

128.708 

141.430 

.3900 

.4286 

104.963 

.3181 

7-3 

.1370 

.20. 

.244 

1.488 

1.780 

129.789 

142.710 

•3933 

•4327 

105.696 

•32°3 

7-4 

•1351 

.202 

.241 

1.492 

1.787 

130.878 

143-979 

.3966 

•4363 

106.420 

•3225 

7-5 

•1333 

.199 

•239 

1.496 

1.794 

131.941 

I45-239 

.3998 

.4401 

107.133 

.3246 

7.6 

.1316 

.197 

•237 

1.500 

i.  80 

I32-995 

146.489 

.4030 

•4439 

107.837 

.3268 

7-7 

.1299 

•195 

•235 

1.504 

1.807 

134.043 

I47.732 

.4062 

•4477 

108.539 

.3289 

7.8 

.1282 

•193 

•233 

1-508 

1.814 

135-063 

148.976 

•4093 

•45J4 

109.219 

•3310 

7-9 

.1266 

.191 

.231 

1.512 

1.821 

136.091 

150.217 

.412^ 

•4552 

109.896 

333° 

8.0 

.1250 

.189 

.228 

1.516 

1.828 

137.110 

151.427 

•4i55 

.4589 

110.565 

335° 

8.1 

.1236 

.188 

.226 

I-5I9 

1-834 

138.111 

152-633 

.4185 

.4625 

111.225 

3370 

8.2 

.1220 

.186 

.224 

1-523 

1.841 

139-093 

153-823 

-4215 

4661 

111.875 

339° 

8.3 

.  I2O5 

.184 

.223 

I-527 

1-847 

140.076 

155.010 

•4245 

4698 

112.522 

34io 

8.4 

.1190 

.l82 

.221 

i-53i 

1.854 

141.060 

156.178 

•4275 

4733 

113-158 

3429 

8-S 

.1176 

.ISO 

.219 

i-534 

1.861 

142.017 

I57-348 

•  43°4 

4768 

113-788 

3448 

8.6 

.1163 

.179 

.217 

I-538 

1.867 

142.974 

158.508 

-4333 

4804 

114.410 

3465 

8-7. 

.1149 

.177 

.215 

I-54I 

1-873 

I43-93I 

159.658 

-4362 

4838 

115.023 

3487 

8.8 

.1136 

.176 

.214 

1-545 

1.879 

44.862 

160.800 

4390 

4873 

II5-633 

35°4 

8.9 

.1124 

.174 

.212 

1-548 

1.885 

45.780 

161.927 

4418 

4906 

16.233 

3522 

9.0 

.mi 

.172 

.210 

1-552 

1.891 

46.700 

63.041 

4446 

4941 

16.827 

3540 

9.1 

.1099 

.171 

.208 

•555 

1.897 

47.627 

64.147 

4474 

4974 

I7-4i5 

355s 

9.2 

.1087 

.170 

.207 

•559 

-903 

48.557 

65.236 

4502 

5007 

117.996 

3576 

9-3 

.  1072 

.168 

.205 

•  562 

.909 

49-554 

66.334 

4532 

5041 

118.571 

3593 

9-4 

.1064 

.l67 

.204 

.565 

•915 

50.312 

67-431 

4555 

5°74 

119.138 

3610 

9-5 

.1058 

.165 

.202 

-569 

.921 

51.188 

68.520 

4582 

5107 

119.702 

3627 

9.6 

.1042 

.l64 

.201 

1.5721.927 

52.066 

69.589 

4609 

5139 

120.259 

3644 

9-7 

.1031 

.162 

.299 

«575I-933 

52.944 

70.650 

4635 

5i7i 

120.810 

3661 

9.8 

.1020 

.l6l 

.298 

•5781-939 

53-794 

71.700 

4661 

5213 

121.355 

3677 

9.9 

.1010 

.160 

.296 

.5821.944 

54-645 

72-754 

4686 

5235 

121.895 

3693 

10.  0 

.  1000 

•159 

•295 

•585I-95° 

55-495 

73-789 

4712 

5266 

122.429 

3710 

74 


NOTES   ON  TABLE   II. 

The  purpose  of  this  table  is  to  determine  the  weight  of  air  compressed 
by  a  machine  of  known  cubic  feet  capacity.  It  is  to  be  used  in  connection 
with  Table  I  for  determining  power  or  work. 

The  barometric  readings  and  elevations  are  made  out  for  a  uniform 
temperature  of  60°  F.  and  are  subject  to  slight  errors  but  not  enough  to 
materially  affect  results.  Table  V  gives  more  accurately  the  relation  be- 
tween elevation  temperature  and  pressure. 


TABLE  II.  — WEIGHTS  OF  FREE  AIR  UNDER  VARIOUS 
CONDITIONS 


Approximate  Baro- 
metric Reading. 
T=6o. 

Atmospheric  Pressure. 

Weight  of  One  Cubic  Foot  at  Given 
Temperature  (Fahr.) 

rt 
DO" 

"  II 

|| 

-20° 

00° 

20° 

40° 

60° 

80° 

100° 

30.52 
30.32 
30.12 

15.0 

14-9 
14.8 

.09211 
.09150 
.09089 

.08811 

.08753 
.08694 

.  08444 
.08388 
•08331 

.08108 
.08054 
.  08OOO 

.07796 
•07744 
.07693 

.07508 
.07458 
.07408 

.07240 

.07192 
.07144 

-600 
—  400 
—  200 

29.91 
29.71 
29.50 

14-7 
14-6 

14-5 

.09027 
.08965 
.08903 

.08635 

.08576 

.08517 

•08275 
.08219 

•07945 
.07895 
.07837 

.07640 
.07589 
•07536 

•  07358, 
.07308 
•07258 

.07095 
.07047 
.06999 

00 
200 
400 

29.30 
29.10 
28.90 

14.4 

14-3 
14.2 

.08842 
.08781 
.08719 

.08458 

.  08400 

.08341 

.08107 
.08050 
.07994 

.07783 
.07729 
•07675 

.07484 
.07432 
•07380 

.07208 
•07158 
.07108 

.06950 

.06902 

.06854 

600 
800 
IOOO 

28.69 
28.49 

28.28 

14-1 
14.0 

13-9 

.08659 
•08597 
•08535 

.08282 

.08224 

.08165 

.07938 
.07882 

.07621 
.07567 
•07513 

.07329 
.07277 
.07225 

.07058 
.07008 
.06957 

.06806 
.06758 
.06709 

I2OO  " 
1400 
IOOO 

28.08 
27.88 
27.67 

13.8 

I3i 
i3.6 

.08474 
.08412 
.08351 

.08106 
.08048 
.07989 

.07769 
.07713 
•07656 

•07459 
•07405 
•07350 

•°7I73 
.07120 
.07068 

.06907 
.06857 
.06807 

.06661 
.06612 

.06564 

1800 
2000 
2IOO 

27-47 
27.27 
27.06 

13-5 
13-4 
13-3 

.08289 
.08228 
.08167 

.07930 

.07871 

.07813 

.07600 

•07544 
.07487 

.07296 
.07242 
.07189 

.07016 
.06965 
•06913 

.06757 
.06707 
•06657 

.06516 

.06468 

.06420 

2300 
2500 
2700 

26.86 
26.66 
26.45 

13.2 
i3.o 

.08106 

.08044 
.07983 

:°7754 
.07695 
.07637 

•07431 
•07375 
.07319 

.07135 

.  07080 
.07026 

.06861 
.  06809 
•06757 

.06607 
•06557 
.06507 

.06371 
.06323 
.06274 

2900 
3100 
33°° 

26.25 
26.05 
25.84 

12.9 

12.8 

12.7 

.07921 

.07860 
.07798 

.07578 
.07518 
.07460 

.07262 
.07206 
.07150 

.06972 
.06918 
.06862 

.06705 
.06652 
.06600 

.06457 
.06407 
•o6357 

.06226 

.06178 
.06130 

35°° 
37°0 
4000 

25.64 
25.44 

12.6 

12.5 

12.4 

•07737 
.07676 
•07615 

.07401 

.07343 
.07284 

.07094 
•07038 
.06981 

.06810 

.06756 
.06702 

•  06549 
.06497 
.06445 

.06307 
•06257 
.06207 

.06082 
•06033 

.05985 

4200 
4400 
4600 

& 


75 


76 


COMPRESSED  AIR 


TABLE   II.  —  Continued. 


25-03 

12.3 

•07553 

.07225 

.06925 

.  06648 

.06393 

.06157 

•05937 

4800 

24.83 

12.2 

.07492 

.07166 

.06868 

.06594 

.06341 

.06107 

.05889 

5000 

24.62 

12.  I 

.07430 

.07108 

.06812 

.06540 

.06289 

.06057 

.05840 

5200 

24.42 

12.0 

.07369 

.  07049 

.06756 

.  06486 

.06237 

.  06007 

•05792 

5400 

24.22 

II.9 

•07307 

.06990 

.  06699 

.06432 

.06185 

•05957 

•05744 

5600 

24*01 

II.  8 

.07246 

.06932 

.  06643 

.06378 

.06133 

.05907 

.05696 

5800 

23.81 

II-7 

.07184 

.06873 

.06587 

.06324 

.06081 

•05857 

.05647 

6100 

23.60 

ii.  6 

.07123 

.06812 

•06530 

.06270 

.06029 

.05807 

•05599 

6300 

23.40 

n-5 

.07061 

•06755 

.06474 

.06216 

•05977 

•05757 

•05551 

6500 

23.20 

11.4 

.07000 

.  06693 

.06418 

.06161 

•05925 

.05707 

.05502 

6800 

22.99 

n-3 

.06938 

.06638 

.06362 

.06108 

•05873 

.05656 

•05454 

7100 

22.79 

II.  2 

.06877 

-06579 

.06305 

.06054 

.05821 

.05606 

.05406 

7300 

22.59 

II.  I 

.06816 

.06520 

.06249 

.  06000 

.05769 

-05556 

•05358 

7600 

22.38 

II.  0 

•06754 

.06462 

.06193 

•05945 

•05717 

.05506 

•05310 

7900 

22.18 

IO-9 

.  06692 

.06403 

.06136 

.05891 

.05665 

•05456 

.05261 

8100 

21.98 

10.8 

.06632 

•  06344 

.06080 

•05837 

•05613 

.05406 

•05213 

8400 

21.77 

10.7 

•06571 

.06285 

.06024 

-05783 

•05561 

•05356 

•05164 

8600 

21.57 

10.6 

.06510 

.06226 

.05968 

.05729 

•05509 

-05306 

.05116 

8900 

21.37 

10.5 

.06448 

.06168 

.05911 

-05675 

•05457 

.05256 

.05068 

9100 

21.  16 

10.4 

.06386 

.06109 

•05855 

.05621 

•05405 

.05206 

.05020 

9400 

20.96 

10.3 

.06325 

.06050 

•05799 

-05567 

•05353 

•05156 

.04972 

9600 

20.76 

10.2 

.06263 

.05991 

•05743 

•05513 

•05301 

.05106 

.04923 

9900 

20.55 

10.  I 

.06202 

•05933 

.05686 

•05459 

.05249 

.05056 

.04875 

IOIOO 

20-35 

IO.O 

.06141 

.05874 

•05630 

•05405 

.05198 

.05006 

.04827 

10400 

20.  15 

9-9 

.06079 

.05816 

•05572 

•05351 

•05146 

.04956 

.04779 

10700 

19.94 

9.8 

.06017 

•05757 

•05517 

•05297 

.05094 

.  04906 

.04730 

1  1  000 

19.74 

9-7 

•05956 

.05698 

.05461 

•05243 

.05041 

.04856 

.04682 

1  1  200 

19-53 

9.6 

.05894 

•05639 

.05404 

.05188 

.04990 

.  04806 

•04633 

11500 

19-33 

9-5 

-05833 

.05580 

•05348 

•05134 

•04937 

.04756 

.04585 

11800 

I9-I3 

9-4 

.05772 

.05522 

.05292 

.05081 

.  04886 

.04706 

•04538 

I2IOO 

18.93 

9-3 

•05711 

•05463 

.06236 

.05027 

.04834 

.04655 

.  04489 

I240O 

l8.72 

9.2 

.05649 

.05404 

•05179 

.04972 

.04782 

.  04605 

.  04440 

12700 

18.52 

9.1 

•05587 

•05345 

.05123 

.04918 

.04730 

•04555 

.04392 

13000 

l8.3I 

9.0 

•05526 

.05286 

.05067 

.04864 

.04678 

.04505 

•04344 

13400 

NOTE    ON    TABLE    III. 

The  table  is  designed  to  compute  readily  weights  of  compressed   air  by 
formula  12,  Art.  8,  viz.,  w  =  —    — .     If  p  is  given  in  pounds   per  square 

inch  the  formula  becomes  w  =  — 

5J.*7XI 

The  value  —  can  most  readily  be  obtained  with  the  slide  rule. 


TABLE   III.  — WEIGHTS   OF   COMPRESSED  AIR 

Pounds  per  Cubic  Foot. 

The  Ratio  -  is  for  absolute  pressure  in  pounds  per  square  inch  and  abso- 
lute temperature  Fahrenheit.     (See  Note  at  foot  of  previous  page.) 


I 

t 

w 

t 

t 

w 

t 

t 

w 

I 

t 

w 

.000 

o  .  oooo 

•255 

.6906* 

".510 

1-3813 

•765 

2.0718 

.005 

•  0135 

.260 

.7041 

•5i5 

1-3947 

.770 

2  .0853 

.010 

.0271 

.265 

^•7177 

.520 

i  .  4083 

•775 

2  .  0988 

.015 

.0406 

.270 

•  7312 

•525 

1.4219 

.780 

2.  1125 

.020 

.0542 

•275 

•7447 

•53° 

1-4355 

•785 

2.  1260 

.025 

.0677 

.280 

•7583 

•535 

i  .  4490 

.790 

2-1395 

.030 

.0813 

.285 

'  -77J9 

•540 

1.4625 

•795 

2.1530 

•°35 

.0948 

.290 

.7852 

•545 

1.4760 

.800 

2.1665 

.040 

.1083 

•295 

.7989 

•55° 

1.4895 

.805 

2.1798 

•045 

.1218 

.300 

•8125 

•555 

1-5030 

.810 

2.1950 

.050 

•1354 

•3°5 

.8260 

.560 

i  .  5  i  66 

•  815 

2.2071 

•°55 

.1489 

.310 

•8395 

•565 

i-5312 

.820 

2.2207 

.060 

.1625 

•3i5 

•8531 

•57° 

1-5437 

•825 

2  •  2343 

•  065 

.  1760 

.320 

.8666 

•575 

I-5572 

.830 

2.2480 

.070 

.1896 

•325 

.8801 

•580 

I-57°7 

•835 

2.2615 

•075 

.2031 

•33° 

•8937 

•585 

1-5843 

.840 

2.2750 

.080 

.2166 

•335 

.9072 

•59° 

i  .  5980 

•  845 

2.2885 

.085 

.2302 

•340 

.9208 

•595 

1.6115 

•  850 

2.3020 

.090 

•  2437 

•345 

•9343 

.600 

1.6250 

•855 

2-3I55 

•095 

•  2573 

•35° 

.9478 

.605 

1-6385 

.860 

2.3290 

.  100 

.2708 

-355 

.9613 

.610 

1.6520 

.865 

2.3425 

.105 

.2843 

.360 

•9749 

.615 

i  .  6654 

.870 

2.3561 

.  no 

.2979 

•365 

.9884 

.620 

1.6792 

•875 

2.3698 

•115 

•3XI4 

•37° 

I  .  O02O 

•  625 

1.6927 

.880 

2.3833 

.  120 

•3250 

•375 

•0155 

.630 

i  .  7062 

.885 

2.3970 

•I25 

•3385 

•380 

.0290 

•635 

1.7198 

.890 

2.4105 

.130 

•3520 

•385 

.0425 

.640 

1-7333 

-895 

2  .4240 

•135 

•3656 

•39° 

.0561 

•645 

1.7468 

.900 

2-4375 

.140 

•3792 

•-395 

.0697 

.650 

i  .  7603 

•905 

2.4510 

•145 

•3927 

.400 

•0833 

•655 

1-7739 

.910 

2  4645 

•15° 

.4062 

•405 

.0968 

.660 

I-7875 

•9i5 

2.4780 

•155 

.4197 

.410 

.1103 

.665 

i  .8010 

.920 

2.4917 

.160 

•4333 

•  415 

.1240 

.670 

1.8145 

•925 

2-5052 

•165 

.4468 

.420 

•1375 

•675 

1.8280 

•93° 

2.5187 

.170 

.4603 

•425 

.1510 

.680 

1.8415 

•935 

2.5323 

•175 

•4739 

•43° 

.1645 

•685 

1-8550 

.940 

2-5459 

.180 

•4875 

•435 

.1780 

.690 

i.  8680 

•945 

2  5594 

•185 

.5010 

.440 

•  J9T7 

•695 

1.8822 

•95° 

2.5730 

.190 

•5145 

•445 

•  2052 

.  700 

1.8959 

•955 

2.5865 

•195 

.5281 

•450 

.2177 

•705 

1.9094 

.960 

2  .  6000 

.200 

.5416 

•455 

•2323 

.710 

1.9229 

•  965 

2  6135 

.205 

•5551 

.460 

•2457 

•7i5 

i-9365 

•97° 

2  .6270 

.210 

.5687 

•465 

•2594 

.720 

1.9500 

•975 

2  •  6405 

•215 

.5822 

.470 

.2730 

•725 

1-9635 

.980 

2.6541 

.  220 

.5958 

•475 

.2865 

•73° 

1.9770 

-985 

2  6670 

.225 

.6094 

.480 

.3000 

•735 

1.9905 

990 

2  6813 

.230 

.6229 

•  485 

•3i35 

.740 

2  .  0042 

•995 

2  .  6949 

•235 

.6364 

•49° 

.3270 

•745 

2.0177 

i  .000 

2  .  7084 

.240 

.6499 

•495 

.3416 

•750 

2.0312 

•245 

-6635 

.500 

•3542 

•755 

2  .  0448 

.2^0 

.6771 

-  =;os 

.3677 

.760 

2  .0582 

77 


TABLE  IV.  *  ~  SPECIAL  TABLE  RELATING  TO  STAGE  COM- 
PRESSION FROM  FREE  AIR  AT  14.7  POUNDS  PRESSURE 
AND  62°  TEMPERATURE. 

Compression  adiabatic  but  cooled  between  stages. 


d 

Single  Stage. 

Two  Stage. 

e 

.9 
( 

1  £ 

ao| 

c 
o 

c  x 

E  °  1 

£ 
ft 

O  ft 

jj 

o  £  •- 

'a  . 

£  fe 

O  fo  •- 

£ 

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jJU 

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r 

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Ti 

H.P. 

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.0197 

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1.68 

.1279 

8.30 

144 

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2.02 

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11.51 

177 

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2.36 

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14.40 

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373 

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32.30 

392 

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405 

.1472 

2.40 

207 

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75 

6.10 

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420 

•1532 

2.47 

214 

.1329 

80 

6.44 

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36-55 

434 

.1590 

2.54 

222 

.1372 

85 

6.78 

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37-90 

447 

•  ^50 

2.60 

227 

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7.12 

.5418 

39.10 

461 

•1705 

2.67 

233 

.1462 

95 

7.46 

.5676 

40.35 

473 

.1758 

2.73 

238 

•  IS00 

100 

7.80 

•5935 

41.65 

485 

.l8l2 

2.79 

242 

•1542 

I05 

8.14 

.6194 

42.30 

497 

.1841 

2.85 

246 

.1578 

no 

8.48 

•6453 

43-75 

508 

.1908 

2.90 

251 

.1615 

IT5 

8.82 

.6712 

45.16 

.1965 

2-99 

256 

.1648 

120 

9.16 

.6971 

46.00 

530 

.2008 

3-02 

259 

.1681 

I25 

9-5° 

.7230 

47-05 

540 

•2045 

3.08 

262 

.  1710 

130 

9.84 

•  7488 

47-80 

550 

.2085 

3-i4 

266 

.1740 

10.18 

•7747 

48.85 

•2135 

3-19 

269 

•1775 

140 

10.52 

.8005 

49.90 

569 

.2176 

3-24 

272 

.1810 

145 

10.86 

.8264 

51  .00 

578 

.2220 

3-29 

276 

-1837 

150 

11.20 

.8522 

5J-70 

587 

•2255 

3-35 

280 

.1865 

*  The  table  is  limited  to  the  special  initial  condition  of  air  specified  in 
the  caption.  The  assumption  of  14.7  as  atmospheric  pressure  makes  the 
weights  and  work  a  little  in  excess  of  average  conditions.  However,  it  is  a 
valuable  and  very  instructive  table. 

78 


PLATES    AND    TABLES 

TABLE    IV  (Continued). 


79 


Two  Stage. 

Three  Stage. 

d 

.9 

'2  o 
n  ^ 

d 

.9 

*$ 

S  "o  | 

o 

d  i 

1^1 

c 

5 

'*-  £' 

ep<  ct 

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**  i 

o^| 

M 

u  ^ 

!*? 

3 

Q 

o  *^ 

o  W 

a  « 

JH      Q^ 

o  w 

4)  5? 

*"*     CX 

1 

I 

"o  ^  » 

ii 

S"*  4J 

o  0  ^ 

g| 

O«  -ij 

1  °  ^ 

K 

Q 

5  ** 

* 

^  *o  -C 

^  $  a> 

•§| 

Hv  —  'C 
o  -^ 

it 

i 

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f  1 

oj  "*"* 

g  pq  a) 

|  P.I 

c  W  w 

1  II 

0 

A 

^ 

t 

M 

(2 

S 

& 

ft 

r 

w 

(r)* 

r, 

H.P. 

^ 

T3 

H.P. 

100 

7.8 

•5936 

2.79 

242 

•1542 

1.98 

I76 

.1450 

150 

II.  2 

.8522 

3-35 

280 

.1865 

2.24 

200 

•1752 

200 

14-6 

I.  IIIO 

3.82 

308 

.2110 

2.44 

215 

.1965 

250 

18.0 

1.3697 

4-24 

332 

•2315 

2.62 

226 

.2140 

300 

21.4 

1.6285 

4-63 

353 

.2490 

2.78 

241 

.2295 

350 

24.8 

1.8872 

4-98 

37° 

.2640 

2.92 

251 

.2418 

4OO 

28.2 

2.1459 

5-31 

386 

.2770 

3-04 

259 

•2535 

450 

31.6 

2  .  4048 

5-6i 

399 

.2895 

3.16 

267 

.2630 

500 

35-o 

2  .  6634 

5-91 

412 

.2915 

3-27 

275 

.2730 

550 

38-4 

2.9221 

3-37 

28l 

.2830 

600 

41.8 

3.l8lO 

3-47 

287 

.2910 

650 

45-2 

3-4395 

3.56 

292 

.2960 

700 

48.6 

3.6982 

3-64 

297 

.3025 

75° 

52.0 

3-9570 

3-73 

302 

.3090 

800 

55-4 

4-2155 

3.80 

307 

•3150 

850 

58.8 

4-4745 

3-83 

312 

.3210 

900 

62.2 

4-7330 

3-96 

3l6 

.3260 

95o 

65.6 

4.9920 

4-03 

320 

.3315 

IOOO 

69.0 

5-2510 

4.  10 

324 

•  3360 

1050 

72.4 

5  •  5°95 

4.17 

328 

.3400 

I  100 

75-8 

5.7684 

4-23 

331 

•3445 

1150 

79-2 

6.0270 

4.29 

334 

•3490 

1200 

82.6 

6-2855 

4-36 

337 

•3525 

1250 

86.0 

6-5445 

4.41 

34i 

•3570 

1300 

89.4 

6  .  8030 

4-47 

344 

•3615 

J35° 

92.8 

7.0620 

4-52 

347 

.3660 

1400 

96.2 

7.3210 

4-58 

35° 

•3685 

145° 

99-6 

7-5795 

4.64 

353 

•3710 

1500 

103.0 

7.8382 

4.70 

356 

•3740 

T55° 

106.4 

8  .  0965 

4-75 

359 

.3780 

1600 

109.8 

8-355° 

4-79 

361 

.3820 

1650 

113.2 

8.6140 

4-83 

363 

•3850 

1700 

116.6 

8.8730 

4.87 

365 

.3880 

175° 

120.0 

9.1320 

4-93 

367 

•3915 

1800 

123.4 

9  .  3900 

4-97 

369 

•3940 

1850 

126.8 

9.6485 

S-02 

•3965 

TABLE  V.  — VARYING  PRESSURES  WITH  ELEVATIONS. 

Solution  of  formula  17,  Art.  17,  viz.  Iogi0/>a  =  1. 16866  — 


122.4 


Elevation  in  Feet. 

Pressure  in  Pounds  per  Square  Inch. 

Temp.  50°  F. 

Temp    35°F. 

Temp.  20°  F. 

0 

14.70 

14.70 

14.70 

IOOO 

14    17 

14.15 

14.14 

2OOO 

13.66 

13-63 

13-99 

3000 

13.  16 

13.12 

13.07 

400O 

12.  69 

1  2  .  63 

12-57 

5000 

12.23 

12  .  l6 

12.09 

5280 

12  .  10 

12.03 

II  .96 

6000 

11.78 

II-7I 

11.63 

7000 

11.36 

11.27 

ii.  18 

8000 

10.95 

10.85 

i°-75 

QOOO 

i°-55 

10.45 

10.33 

1  0000 

10.  17 

10.06 

9.94 

12500 

9.28 

9-15 

9.02 

15000 

8.46 

8.3, 

8.18 

TABLE  VI.*— HIGHEST  LIMIT  TO  EFFICIENCY  WHEN 
COMPRESSED  AIR  IS  USED  WITHOUT  EXPANSION, 
ASSUMING  ATMOSPHERIC  PRESSURE  =  14.5  POUNDS 
PER  SQUARE  INCH. 


r 

h 

E 

r 

h 

E 

r 

h 

E 

I  .2 

6.66 

91.4 

5-2 

140.0 

49.0 

9-2 

273-3 

40.2 

1.4 

13-3 

84.9 

5-4 

146.6 

48.3 

9-4 

280.0 

39-9 

1.6 

20.0 

79.8 

5-6 

153-3 

47-7 

9.6 

286.6 

39-6 

1.8 

26.6 

75-6 

5-8 

1  60.0 

47-0 

9-8 

293  3 

39-3 

2  .O 

33-3 

72.0 

6.0 

166.6 

46.5 

10.  0 

300.0 

39-o 

2.2 

40.0 

69.2 

6.2 

173-3 

46.0 

10.25 

308.3 

38.6 

2.4 

46.6 

66.7 

6.4 

180.0 

45-5 

10.50 

316.6 

38.5 

2.6 

53-3 

61.9 

6.6 

186.6 

45-o 

IO-75 

325-0 

38.0 

2.8 

60.0 

62.4 

6.8 

193-3 

44-5 

11.00 

333-3 

37-9 

3-o 

66.6 

60.7 

7.0 

200.0 

44-o 

11.25 

341-6 

37-7 

3-2 

73-3 

59-i 

7.2' 

206.  6 

43-6 

1  1  .50 

350-o 

37-4 

3-4 

80.0 

57-8 

7-4 

213-3 

43-i 

n-75 

353  3 

37-i 

3-6 

86.6 

56-4 

7.6 

220  .0 

42.8 

12.00 

366.6 

36  9 

3-8 

93-3 

SS-2 

7-8 

226  6 

42.4 

12.25 

375-o 

36.7 

4.0 

100  .0 

54-1 

8.0 

233  •  3 

42  o 

I2.5C 

383-3 

36.4 

4.2 

1  06  6 

53-i 

8.2 

240.0 

41.7 

12-75 

391.6 

36.2 

4-4 

ii3-3 

52.1 

8.4 

246.  6 

41.4 

13.0 

400.0 

36.0 

4-6 

120.0 

5i-3 

8.6 

253-3 

41.1 

14.0 

433-3 

35-2 

4.8 

126.6 

50-5 

8.8 

260.0 

40.8 

15.0 

466.6 

34-5 

5-o 

133-3 

49-7 

9.0 

266.6 

40-5 

16.0 

500.0 

33-8 

*  This  table  reveals  the  limit  of  efficiency  when  air  is  applied  without 
utilizing  any  of  its  expansive  energy. 

The  column  headed  r  gives  the  ratio  of  compression,  while  that  headed  h 
gives  the  water  head  equivalent  to  a  pressure  given  by  the  ratio  r  on  the 
assumption  that  one  atmosphere  is  a  pressure  of  14.5  pounds  per  square 
inch  or  a  water  head  of  33.3  feet,  this  being  more  nearly  the  average  condi- 
tion than  14.7,  which  is  so  commonly  taken. 

It  should  be  understood  that  this  efficiency  cannot  be  reached  in  practice, 
—  it  being  reduced  by  friction  of  air  and  machinery  and  by  clearance  in 
any  form  of  engine. 

80 


PLATES  AND  TABLES 


81 


TABLE   VII.  —  EFFICIENCY  OF  DIRECT  HYDRAULIC  AIR 
COMPRESSORS. 


Formula  26,  Art.  25,   viz.  E 


2'3 


Water  Head. 

Gage  Pressure. 

Absolute  Pres- 
sure.^ 

Atmospheres 

=  r 

Efficiency, 
E 

0.0 

0.0 

14-5 

i 

1  .00 

33-3 

14.5 

29.0 

2 

.69 

66.6 

29.0 

43-5 

3 

•55 

100.  0 

43-5 

58.0 

4 

.46 

.  133-3 

58.0 

72-5 

5 

.40 

166.6 

72.5 

87.0 

6 

•36 

200.0 

87.0 

101.5 

7 

•33 

233-3 

101.5 

116.0 

8 

•3° 

266.0 

116.0 

i3°-5 

9 

.28 

300.0 

130-5 

M5-° 

10 

.26 

TABLE   VIII.  —  COEFFICIENT   "c»   FOR  VARIOUS   HEADS 
AND  DIAMETERS. 


! 

d" 

i  =  i  " 

i=  2" 

i=3/r 

*=4" 

f-s" 

ft 

0.603 

0.606 

0.610 

0.613 

0^616 

\ 

0.602 

0.605' 

0.608 

o.  610 

0.613 

i 

0.601 

0.603 

0.605 

o.6o6_ 

0.607 

0 

0.60  1 

o.  60  1 

o.  602 

0.603 

0.603 

o.  600 

0.600 

0.600 

0.600 

0.600 

^ 

0-599 

0-599 

o-599 

0.598 

0.598 

3 

o  599 

0.598 

0-597 

0-596  ' 

o  596 

3* 

o  599 

o  597 

0.596 

0-595 

0.594 

4 

0.598 

o  597 

o  595 

0-594 

o-593 

4i 

o  598 

0.596 

0.596 

o-593 

0.592 

Table  VIII  gives  the  experimental  coefficients  for  orifices  for  determining 
the  weight  of  air  passing  by  formula: 

Weight  (0  =  0.6299  c 

Q  =  Weight  of  air  passing  in  pounds  per  second. 
c  =  Experimental  coefficient. 
d  =  Diameter  of  orifice  in  inches. 
i  =  Difference  of  pressure   inside   and  outside  of  orifice  in  inches  of 

water. 
/  =  Absolute  temperature  of  air  back  of  orifice. 


82 


COMPRESSED  AIR 


TABLE   IX.— FRICTION  IN  AIR  PIPES. 


Divide  the  number  corresponding  to  the  diameter  and  volume 

"o  o. 

4J           S-H 

by  the  ratio  of  compression.     The  result  is  the  loss  in  pounds  per 
square  inch  in  1000  feet  of  pipe. 

8  <i  £ 

Diameter  of  Pipe  in  Inches. 

3 
0 

i 

1 

i 

'i 

i* 

If 

2 

.* 

3 

6 

27-3 

35-4 

-83 

.26 

12 

108.3 

14.26 

3-32 

1.05 

24 

56.64 

13.28 

4.20 

1.71 

.78 

36 

126.4 

29.86 

9-45 

3-84 

i-75 

48 

226.6 

53-15 

16.80 

6.83 

3.12 

I.  60 

60 

84.94 

26.26 

10.  70 

4.87 

2.50 

72 

II9.8 

37-90 

15.40 

7-03 

3.62 

I.I7 

84 

163.7 

51.46 

20.90 

9-55 

4.91 

i-59 

96 

67.21 

27.30 

12.48 

6.41 

2.07 

108 

85.06 

34-55 

15.80 

8.12 

2.62 

120 

105.0 

42.67 

19.50 

10.  OO 

3-25 

1S° 

164.  i 

66-53 

30-47 

15.66 

5-o6 

I-85 

1  80 

96.00 

43-87 

22.54 

7.28 

2.67 

210 

130-7 

59-7i 

30.70 

9.91 

3-63 

if 

2 

•i 

3 

si 

4 

4i 

5 

6 

240 

78.00 

40.09 

12.94 

4-74 

2.13 

270 

98.70 

50.72 

16.48 

6.00 

2.70 

300 

121.  8 

62.62 

20.23 

7.41 

3-33 

330 

75.78 

24-57 

8.97 

4-03 

360 

90.29 

29.  12 

10.67 

4.8c 

390 

105-5 

34.20 

12-53 

5-63 

420 

122.8 

39.64 

14.52 

6-53 

2.87 

45° 

45-5° 

16.67 

7-49 

3-30 

480 

51.88 

18.97 

8-53 

3-75 

510 

58.44 

21.42 

9.62 

4-23 

540 

65-39 

24.01 

10.79 

4-75 

570 

73.00 

26.75 

12  .02 

5-29 

2.94 

600 

80.90 

29.64 

I3-32 

5-86 

3-25 

660 

97.90 

35.87 

16.  12 

7.09 

3-93 

720 

116.50 

42.68 

19.19 

8-43 

4.68 

780 

50.  10 

22.50 

9.00 

5-5° 

3-25 

840 

58.10 

26.  II 

11.48 

6-37 

3-76 

900 

66.  70 

29.98 

13.18 

7-32 

4-32 

960 

75-88 

34.10 

15.00 

8.32 

4.92 

1020 

85-65 

38.50 

16.93 

9.40 

5-55 

2-23 

1080 

96.04 

43-17 

18.98 

10.53 

6.22 

2.50 

II4O 

107.00 

48.10 

21.15 

n-74 

6-93 

2.79 

1200 

53-29 

23-44 

13.01 

7.68 

3-09 

1320 

64.49 

28.36 

IS-74 

9.29 

3-73 

1440 

76.74 

33-75 

18.73 

1  1.  06 

4-44 

1560 

90.05 

39.61 

2  I  .  89 

12.98 

5-22 

1680 

104.45 

45-95 

25-50 

16.78 

6.05 

TABLE  IX  (Continued}. 


|3f 

fc    a,    G 

.a  S  jg 
•°  fe  ? 

<5  *o  a 

Diameter  of  Pipe  in  Inches 

4 

4* 

5 

6 

8 

I  O 

I  2 

1800 

52-73 

28.23 

17.82 

6-95 

1-65 

1920 

6o.OO 

33-3° 

19.66 

7.90 

1.87 

2040 

67.74 

37-59 

22.20 

^8.92 

2.12 

2160 

75-94 

42.15 

24.89 

IO.OO 

2-37 

2280 

84.60 

46.95 

27.65 

n-*4 

2.64 

2400 

93-74 

52.02 

>.72 

12-35 

2-93 

0.96 

2520 

53.38 

33-87 

13.61 

3-23 

1.  06 

2640 

62.96 

37-17 

14.94 

3-55 

.16 

2780 

68.  8  1 

40.66 

16.33 

3-88 

.27 

2880 

74.92 

44.78 

17.78 

4.22 

•38 

3000 

81.30 

48.00 

19.29 

4-58 

.50 

33°o 

98.37 

58.08 

23-34 

5-54 

.81 

3600 

69.13 

27-78 

6-59 

2.  l6 

0.87 

3900 

81.13 

32.60 

7-74 

2-53 

1.02 

4200 

94.09 

37.8i 

8.97 

2.94 

1.18 

4500 

43-41 

10.30 

3-37 

1.36 

4800 

49-39 

11.72 

3-84 

1.54 

5100 

55.76 

13-23 

4-34 

1.74 

5400 

62.51 

14.83 

4.86 

1.95 

5700 

69.62 

16.53 

5-41 

2.18 

6000 

77-18 

18.31 

6.00 

2.41 

6600 

22.  l6 

7.26 

2.92 

7200 

26.37 

8.64 

3-47 

7800 

30.95 

10.10 

4.07 

8400 

35-90 

11.76 

4-73 

9000 

41  .20 

13.50 

5.40 

9600 

46.88 

15.36 

6.17 

10200 

52.92 

17.34 

6.97 

10800 

59.36 

19.44 

7.81 

II400 

66.11 

21.66 

8.70 

12000 

73.25 

24.00 

9-64 

13200 

29.04 

11.67 

I440O 

34.56 

13.89 

15600 

40.56 

16.30 

16800 

47.40 

18.90 

18000 

54.00 

21.70 

I92OO 

61.43 

24.70 

20400 

69.36 

27.87 

2IOOO 

77-75 

31-25 

22800 

86.64 

34-82 

24OOO 

96.00 

38-58 

This  table  represents  the  author's  formula  20,  Chap.  IV. 

l    Va* 


f  =  Loss  of  pressure  in  pounds  per  square  inch. 
c  ==•  An  experimental  coefficient. 
/  =  Length  of  pipe  in  feet. 
d  =  Diameter  of  pipe  in  inches. 
va—  Cubic  feet  of  free  air  passing  per  second. 
r  =  Ratio  of  compression  from  free  air. 

83 


COMPRESSED  AIR 


§       8       3 


PLATES    AND    TABLES 


8       8       S        5       §        § 


I 

^      CL,    co 


COMPRESSED  AIR 


3           «           oo           i-           co           o 
i~~ 

_____      _           _        5  _ 

•»«!_«             ^ 

TF""~  ^ 

:--! 

[J4liiii  I-        § 

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V                                        -  5 

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_  .  \   _  _                                                -  -     rH 

::i::                  ::  «5 
::::::!::5::::!::::::::|S 

L 

-    i                                2 

____                                                                          -            _            _                             _                           ^     

_  _       _  _  S  _ 

3                                              rH     ft 

.                                            ^   -                                 -     g    ^ 

S   -                                           ~\       ~                         '-      1H     ® 

S  "              "~    "  "' 

§  £ 

tSi 

^ 

\                           L.                             o> 

q             *  _, 

k                        4                             o> 

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c    ft                                       »  "  " 

0    ^ 

02      03                                                                            5 

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a  .       :  :  ;  :;.::_ 

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a  t-> 

oO 

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^  t     \ 

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_L.  Jt   .      g 

\n]    g 

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..;  :  :  :.  g 

T  8 

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S        S        g        9 


PLATES    AND    TABLES 


87 


§       g       S       g        8        § 


• 

<9t 

CO       . 


W       Q,   3 

H    vS""" 


S        S 


88 


COMPRESSED  AIR 


§       8 


TABLE   X.— TABLE   OF   CONTENTS   OF   PIPES  IN   CUBIC 
FEET  AND  IN  U.  S.  GALLON. 


Diam. 

For  i  Foot 

in  Length. 

Dicim  . 

For  i  Foot 

in  Length. 

Diam. 
in 
[nches. 

in  Deci- 
mals of 
a  Foot. 

Cubic  Feet. 
Also  Area 
in  Square 
Feet. 

Gallons  of 
23  i  Cubic 
Inches. 

Diam. 
in 
[nches. 

in  Deci- 
mals of 
a  Foot. 

Cubic  Feet. 
Also  Area 
in  Square 
Feet. 

Gallons  of 
23  i  Cubic 
Inches. 

i 

.0208 

.0003 

.0026    '> 

II. 

.9167 

.6600 

4-937 

A 

.0260 

.0005 

.0040 

4    * 

•9375 

.6903 

5-163 

I 

•0313 

.0008 

,,.0057 

•9583 

•7213 

5-395 

A 

•0365 

.0010 

.0078 

1 

•9792 

•7530 

5-633 

* 

.0417 

.0014 

.0102 

12. 

Foot 

•7854 

5.876 

& 

.0469 

.0017 

.0129 

1 

.042 

-8523 

6-375 

I 

.0521 

.0021 

^0*59 

13- 

.083 

.9218 

6.895 

if 

•0573 

.0026 

.0193 

i 

•125 

.9940 

7-435 

a 

4 

.0625 

.0031 

.0230 

14. 

.167 

.069 

7-997 

if 

.0677 

.0036 

.0270 

\ 

.208 

.147 

8.578 

1 

.0729 

.O042 

.0312 

*s- 

•250 

.227 

9.180 

I  M 

.0781 

.0048 

•°359 

\ 

.292 

.310 

9.801 

•0833 

•°°55 

.0408 

16. 

•333 

•396 

10.44 

'  i 

.  1042 

.0085 

.0638 

\ 

•375 

•485 

ii.  ii 

i 

.1250 

.0123 

.0918 

i7- 

.417 

•576 

11.79 

I 

.1458 

.0168 

.1250 

\ 

.458 

.670 

12.50 

2  . 

.1667 

.0218 

.1632 

18. 

.500 

.767 

13.22 

i 

•1875 

.0276 

.2066 

\ 

•542 

.867 

J3-97 

i 

.2083 

.0341 

•255o 

19. 

•583 

.969 

J4-73 

S 

.2292 

.0413 

•3085 

i 

.625 

2.074 

I5-52 

3- 

.2500 

.0491 

•3673 

20. 

.666 

2.182 

16.32 

.2708 

.0576 

.4310 

\ 

.708 

2.292 

i7-i5 

1 

.2917 

.0668 

.4998 

21. 

•75° 

2.405 

17.99 

§ 

•3I25 

.0767 

•5738 

i 

.792 

2.521 

18.86 

4- 

•3333 

.0873 

.6528 

22. 

•833 

2.640 

J9-75 

i 

•3542 

•  0985 

•7370 

i 

•875 

2.761 

20.65 

) 

•3750 

•  II05 

.8263 

23- 

.917 

2.885 

22.58 

1 

.3958 

.1231 

.9205 

i 

•958 

3.012 

21-53 

5- 

.4167 

.1364 

i.  020 

24. 

2.000 

3.142 

23.5° 

i 

•4375 

•i5°3 

1.  124 

25- 

2.083 

3.409 

25-50 

i 

•4583 

•  J^SQ 

1-234 

26. 

2.166 

3.687 

27-58 

* 

.4792 

.1803 

1-349 

27. 

2.250 

3-976 

29.74 

6. 

.5000 

.1963 

.469 

28. 

2-333 

4.276 

3i-99 

i 

.5208 

.2130 

•594 

29. 

2.416 

4.587 

34-31 

i 

•54i7 

•2305 

•724 

30. 

2.500 

4.909 

36.72 

1 

•5625 

•  2485 

•859 

31- 

2-583 

5.241 

39.21 

7 

•5833 

.2673 

•     .999 

32- 

2.666 

5-585 

41.78 

i 

.6042 

.2868 

2.144 

33- 

2.750 

5-940 

44-43 

i 

.6250 

.3068 

2-295 

34 

2-833 

6-305 

47-17 

i 

.6458 

•3275 

2.450 

35- 

2.916 

6.68! 

49-98 

8. 

.6667 

•  349° 

2.611 

36. 

3.000 

7.069 

52.88 

i 

.6875 

•37i3 

2-777 

37- 

3-083 

7.468 

55-86 

^ 

.7083 

•  3940 

2.948 

38. 

3.166 

7.876 

58.92 

4 

.7292 

•4i75 

3-125 

39- 

3-250 

8.296 

62.06 

9- 

.7500 

.4418 

3-3°5 

40. 

3-333 

8.728 

65.29 

i 

.7708 

.4668 

3-492 

41. 

3.416 

9.168 

68.58 

* 

.7917 

•4923 

3.682 

42. 

3-5oo 

9.620 

71.96 

I 

.8125 

•5185 

3.879 

43- 

3-583 

10.084 

75-43 

10. 

•8333 

•5455 

4.081 

44- 

3.666 

10.560 

79.00 

i 

.8542 

•573° 

4.286 

45- 

3-75° 

11.044 

82.62 

i 

.8750 

.6013 

4.498 

46. 

3-833 

i  i  .  540 

86.32 

I 

.8958 

•6303 

4.714 

47- 

3.916 

12.048 

90.12 

48. 

4.000 

12.566 

94.02 

8g 


TABLE  XI  — CYLINDRICAL    VESSELS,    TANKS,    CISTERNS, 

ETC. 

Diameter  in  Feet  and  Inches,  Area  in  Square   Feet,  and  U.  S.  Gallons 
Capacity  for  One  Foot  in  Depth. 

i  cubic  foot 

i  gallon  =  231  cubic  inches  = —     — =  0.13368  cubic  feet. 

7.4805 


Diam. 

Area. 

Gals. 

Diam. 

Area. 

Gals. 

Diam. 

Area. 

Gals. 

Ft.    In. 

Sq.  Ft. 

i  Ft. 
Depth. 

Ft.    In. 

Sq.  Ft. 

i  Ft. 
Depth. 

Ft.    In. 

Sq.  Ft. 

i  Ft. 
Depth. 

.785 

5.89 

5      5 

23-04 

172.38 

I7      6 

240.53 

1799-3 

I 

.922 

6.89 

5     6 

23.76 

177.72 

17     9 

247-45 

1851.1 

2 

.069 

8.00 

5     7 

24.48 

183-15 

18 

254.47 

1903.6 

3 

.227 

9.18 

5     8 

25-22 

188.66 

18     3 

261.59 

1956.8 

4 

.396 

10.44 

5     9 

25-97 

194.25 

18     6 

268.80 

2010.8 

5 

•576 

n-79 

5  1° 

26.73 

199.92 

18     9 

276.12 

2065.5 

6 

.767 

13.22 

5  ii 

27-49 

205.67 

19 

283.53 

2120.9 

7 

.969 

!4-73 

6 

28.27 

211  .51 

!9     3 

291.04 

2177.1 

8 

2.182 

16.32 

6    3 

30.68 

229.50 

19     6 

298.65 

2234.0 

9 

2.405 

17.99 

6    6 

33-18 

248.23 

19     9 

306.35 

2291.7 

10 

2.640 

19-75 

6     9 

35-78 

267.69 

20 

314.16 

2350-I 

ii 

2.885 

21.58 

7 

38.48 

287.88 

20     3 

322.06 

2409.2 

2 

3-142 

23-5° 

7     3 

41.28 

308.81 

20       6 

330.06 

2469.1 

2        I 

3-409 

25-5° 

7     6 

44.18 

330-48 

20     9 

338.16 

2529-6 

2       2 

3.687 

27-58 

7     9 

47-17 

352.88 

21 

346.36 

2591.0 

2       3 

3-976 

29.74 

8 

50.27 

376.01 

21     3 

354-66 

2653.0 

2       4 

4.276 

3!-99 

8     3 

53-46 

399-88 

21        6 

363.05 

2715.8 

2     5 

4.587 

34-31 

8     6 

56.75 

424.48 

21     9 

371-54 

2779-3 

2       6 

4.909 

36.72 

8     9 

60.13 

449.82 

22 

38o.I3 

2843.6 

2     7 

5-241 

39-21 

9 

63.62 

475-89 

22       3 

388.82 

2908.6 

2       8 

5.585 

41.78 

9     3 

67.20 

502-7° 

22       6 

397-61 

2974-3 

2       Q 

5-940 

44-43 

9     6 

70.88 

530.24 

22       9 

406.49 

3040  .  8 

2     10 

6-3°5 

47.16 

9     9 

74.66 

558.51 

23 

4I5-48 

3108.0 

211 

6.681 

49-98 

10 

78.54 

587-52 

23       3 

424.56 

3I75-9 

3 

7.069 

52.88 

10     3 

82.52 

617.26 

23     6 

433-74 

3244.6 

3     i 

7-467 

55-86 

10     6 

86.59 

647-74 

23     9 

443-01 

33M.O 

3     2 

7.876 

58.92 

10     9 

90.76 

678.95 

24 

452.39 

3384.1 

3     3 

8.296 

62.06 

ii 

95-03 

710.90 

24     3 

461.86 

3455-o 

3     4 

8.727 

65.28 

ii     3 

99.40 

743-58 

24     6 

47L44 

3526.6 

3     5 

9.168 

68.58 

ii     6 

103.87 

776.99 

24     9 

481  .  ii 

3598.9 

3     6 

9.621 

71-97 

ii     9 

108.43 

811  .  14 

25 

490.87 

3672.0 

3     7 

10.085 

75-44 

12 

113-10 

846  .  03 

25     3 

500.74 

3745-8 

3     8 

10-559 

78.99 

12     3 

117.86 

881  .  65 

25     6 

510.71 

3820.3 

3     9 

i  i  .  045 

82.62 

12       6 

122.72 

918.00 

25     9 

520.77 

3895-6 

3  10 

11.541 

86.33 

12     9 

127.68 

955-09 

26 

530.93 

3971.6 

3  ii 

I  2  .  048 

90.13 

13 

132.73 

992.91 

26     3 

54I.I9 

4048.4 

4 

12.566 

94.00 

13     3 

137.89 

io3i-5 

26     6 

551-55 

4125.9 

4      i 

I3-095 

97.96 

13     6 

143-14 

1070.8 

26     9 

562.00 

4204.  i 

4     2 

I3-635 

102.00 

i3     9 

148.49 

i  i  10.  8 

27 

572.56 

4283.0 

4     3 

I4.I86 

106.12 

14 

153-94 

ii5i-5 

27     3 

583-21 

4362  .  7 

4     4 

14.748 

110.32 

i4     3 

159.48 

1193.0 

27     6 

593-96 

4443-1 

4    5 

I5-32I 

114.61 

14     6 

165.13 

1235-3 

27     9 

604.81 

4524.3 

4     6 

15.90 

118.97 

14    9 

170.87 

1278.2 

28 

6i5.75 

4606.2 

4     7 

16.50 

123.42 

15 

176.71 

1321-9 

28     3 

626.80 

4688.8 

4     8 

17.10 

127-95 

iS     3 

!82.65 

1366.4 

28     6 

637-94 

4772.1 

4     9 

17.72 

132.56 

15     6 

188.69 

1411.5 

28     9 

649.  18 

4856.2 

4  10 

18.35 

137-25 

i5     9 

194-83 

1457-4 

29 

660.52 

4941.0 

4  ii 

18.99 

142.02 

16 

2OI  .06 

1504-1 

29     3 

671  .96 

5026.6 

5 

19.63 

146.88 

16     3 

207.39 

I55L4 

29     6 

683.49 

5H2.9 

5     i 

2O.29 

151.82 

16     6 

213.82 

1599-5 

29     9 

695-13 

5I99-9 

5     2 

20.97 

156.83 

16    9 

220.35 

1648.4 

30 

706.86 

5287.7 

5     3 

21  .65 

161.93 

17 

226.98 

1697-9 

S      4 

22.34 

167.12 

i?     3 

233-7J 

1748.2 

90 


PLATES    AND    TABLES 


91 


TABLE  XII.  —  STANDARD  DIMENSIONS  OF  WROUGHT-IRON 
WELDED   PIPE. 

(National  Tube*<Works.) 


Nominal 
Inside 
Diameter 

Actual 
Outside 
Diameter. 

Actual 
Inside 
Diameter. 

Internal  Area. 

External   Area. 

Ins. 

Ins. 

Ins. 

Sq.'  In. 

Sq.  Ft. 

Sq.  In. 

Sq.  Ft. 

i 

•405 

.270 

•057 

.OOO4 

.1288 

.0009 

| 

•540 

*   -364 

.104 

.0007 

.2290 

.0016 

f 

•675 

•493 

.191 

.0013 

•3578 

.0025 

\ 

.840 

.622 

.304 

.0021 

-554 

.0038 

I 

1.050 

.824 

•533 

.0037 

.866 

.0060 

I 

'•3*5 

1.048 

.861 

.0060 

I-358 

.0094 

il 

i.  660 

1.380 

1.496 

.OIO4 

2.164 

.0150 

ri 

1.900 

1.610 

2.036 

.0141 

2-835 

•0197 

2 

2-375 

2.067 

3-356 

•0233 

4-43° 

.0308 

2* 

2.875 

2.468 

4.780 

•0332 

6.492 

.0451 

3 

3-5°° 

3.067 

7-383 

•0513 

9.621 

.0668 

si 

4.000 

3.548 

9.887 

.0689 

12,566 

•0875 

4 

4.500 

4.026 

12.730 

.0884 

15.904 

.1104 

4* 

5  .000 

4.508 

15.96! 

.1108 

19-635 

•1364 

5 

5-563 

5-°45 

19.986 

.1388 

24.301 

.1688 

6 

6.625 

6.065 

28.890 

.2006 

34.472 

•2394 

7 

7.625 

7.023 

38.738 

.2690 

45  •  664 

•3171 

8 

8.625 

7.981 

50.027 

•3474 

58.426 

•4057 

9 

9.625 

8-937 

62.730 

•4356 

72.760 

•5053 

10 

10.75 

10.018 

78-823 

•5474 

90.763 

-6303 

ii 

u-75 

I  I  .  000 

95-033 

.6600 

108.434 

•7530 

12 

12.75 

I  2  .  000 

113.098 

•7854 

127.677 

.8867 

13 

14 

I3-25 

137.887 

•9577 

I53-938 

i  .  0690 

14 

15 

14.25 

159-485 

1.1075 

176.715 

1.2272 

15 

16 

l5-25 

182.665 

1.2685 

201.062 

I-3963 

I? 

18 

I7-25 

239.706 

1.6229 

254.470 

1.7671 

19 

20 

19.25 

291.040 

2.02II 

3I4-i59 

2.1817 

21 

22 

21.25 

354.657 

2.4629 

380.134 

2  .  6398 

23 

24 

23-25 

424-558 

2.9483 

452.390 

3.I4I6 

TABLE  XIII.  — HYPERBOLIC   LOGARITHMS. 


N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

.OI 

.00995 

1-57 

.45108 

2.13 

.75612 

2.69 

•98954 

.02 

.01980 

1.58 

•45742 

2.14 

.76081 

2.70 

•99325 

•03 

.02956 

1.59 

.46373 

2.15 

•76547 

2.71 

•99695 

.04 

.03922 

i.  60 

.47000 

2.16 

.77011 

2.72 

i  .  00063 

•05 

.04879 

1.61 

•47623 

2.17 

•77473 

2-73 

i  .  00430 

.06 

.05827 

1.62 

.48243 

2.18 

•77932 

2.74 

.00796 

.07 

.06766 

1.63 

.48858 

2.19 

•78390 

2.75 

.01160 

.08 

.07696 

1.64 

.49470 

2.20 

.78846 

2.76 

•01523 

.OQ 

.08618 

1.65 

.50078 

2.21 

.79299 

2.77 

.01885 

.10 

.09531 

1.66 

.50681 

2.22 

•79751 

2.78 

.02245 

.11 

•  I0436 

1.67 

.51282 

2.23 

.  80200 

2-79 

.02604 

.12 

•II333 

1.68 

•5l879 

2.24 

.80648 

2.80 

.02962 

•13 

.  12222 

1.69 

.52473 

2.25 

.81093 

2.81 

•03318 

.14 

.13103 

1.70 

•53063 

2.26 

.81536 

2.82 

•03674 

•15 

•13977 

•71 

•53649 

2.27 

.81978 

2.83 

.04028 

.16 

.  14842 

.72 

•54232 

2.28 

.82418 

2.84 

.  04380 

•17 

.15700 

•73 

.54812 

2.29 

•82855 

2.85 

•04732 

.18 

•I655I 

•74 

.55389 

2.30 

.83291 

2.86 

.05082 

.19 

•17395 

•75 

.55962 

2.31 

.83725 

2.87 

•05431 

.20 

.18232 

.76 

•56531 

2.32 

•84157 

2.88 

.05779 

.21 

.  19062 

•77 

.57098 

2.33 

.84587 

2.89 

.06126 

.22 

.19885 

.78 

.57661 

2-34 

•85015 

2.90 

.06471 

•23 

.2O70I 

•79 

.58222 

2-35 

•85442 

2.91 

.06815 

.24 

.21511 

.80 

•58779 

2.36 

.85866 

2.92 

.07158 

•25 

.22314 

.81 

•59333 

2-37 

.86289 

2-93 

.07500 

.26 

.23111 

.82 

•59884 

2.38 

.86710 

2.94 

.07841 

.27 

.  23902 

.83 

.60432 

2-39 

.87129 

2-95 

.08181 

.28 

.24686 

.84 

.60977 

2.40 

•87547 

2.96 

•08519 

.29 

.25464 

.85 

•61519 

2.41 

.87963 

2.97 

.08856 

1.30 

.26236 

.86 

.62058 

2.42 

.88377 

2.98 

.09192 

I-3I 

.27003 

.87 

•62594 

2-43 

.88789 

2-99 

.09527 

1.32 

.27763 

1.88 

.63127 

2.44 

.89200 

3-oo 

.09861 

1-33 

.28518 

1.89 

.63658 

2-45 

.89609 

3.01 

.  10194 

1-34 

.29267 

1.90 

.64185 

2.46 

.90016 

3.02 

.10526 

1-35 

.30010 

1.91 

.64710 

2-47 

.90422 

3-03 

.10856 

1.36 

.30748 

1.92 

•65233 

2.48 

.90826 

3-04 

.11186 

•37 

.31481 

1-93 

•65752 

2.49 

.91228 

3.05 

.11514 

.38 

.32208 

1.94 

.66269 

2.50 

.91629 

3.06 

.  11841 

•39 

.32930 

1-95 

•66783 

2.51 

.92028 

3.07 

.12168 

.40 

.33647 

1.96 

.67294 

2.52 

.92426 

3-o8 

•12493 

.41 

•34359 

1-97 

.67803 

2-53 

.92822 

3-09 

.12817 

.42 

.35066 

1.98 

.68310 

2-54 

.93216 

3.10 

.13140 

•43 

.35767 

1.99 

.68813 

2-55 

.93609 

3-ii 

.13462 

.44 

•  36464 

2.00 

•69315 

2.56 

.94001 

3.12 

•13783 

•45 

.37156 

2.01 

.69813 

2.57 

•94391 

3-13 

.14103 

•46 

.37844 

2.02 

..7031° 

2.58 

•94779 

3-14 

.14422 

•47 

.38526 

2.03 

.  70804 

2.59 

.95166 

3-15 

•  14740 

.48 

.39204 

2.04 

•7i295 

2.60 

•95551 

3-i6 

•15057 

.49 

.39878 

2.05 

.71784 

2.61 

•95935 

3-17 

•15373 

•50 

•40547 

2.06 

.72271 

2.62 

•96317 

3-18 

.15688 

•51 

.41211 

2.07 

.72755 

2.63 

.96698 

3-19 

.16002 

•52 

.41871 

2.08 

•73237 

2.64 

.97078 

3-20 

•16315 

•53 

.42527 

2.09 

•737i6 

2.65 

•97454 

3.21 

.16627 

•54 

.43178 

2.10 

.74194 

2.66 

.97833 

3-22 

.16938 

•55 
.56 

.43825 
.  44469 

2.  II 
2.12 

.  74669 
•75i42 

HI 

.98208 
.98582 

3-23 
3-24 

i.  17248 
I-I7557 

92 


TABLE  XIII.    Continued.  —  HYPERBOLIC   LOGARITHMS. 


N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

3-25 

I   17865 

3.81 

I.33763 

4-37 

1.47476 

4-93 

1-59534 

3.26 

i8i73 

3.82 

1-34025 

4.38 

I-477°5 

4-94 

1  -59737 

3.27 

18479 

3.83 

i   34286 

4-39 

1-47933 

4-95 

i  •  59939 

3.28 

18784 

3-84 

1  -34547,1 

4.40 

1.48160 

4.96 

i  .  60141 

3-2Q 

19089 

3-85 

i  .  34807 

4-41 

1.48387 

4-97 

1.60342 

3-30 

19392 

3-86 

1-35067 

*4-42 

1.48614 

4.98 

1.60543 

3.31 

19^5 

3-87 

^•35325 

4-43 

i  .  48840 

4.99 

1.60744 

3-32 

19996 

3-88 

I-35584 

4-44 

i  .  49065 

5-00 

i  .  60944 

3-33 

20297 

3.89 

1.35^41 

4-45 

1.49290 

S.oi 

i  .61144 

3-34 

20597 

3.90 

1.36093 

4.46 

I-495I5 

5-02 

1-61343 

3-35 

20896 

3.9i 

t-36354 

4-47 

1-49739 

5-03 

1.61542 

3-36 

21194 

3.92 

i  .  36609 

4.48 

i  .  49962 

5.04 

1.61741 

3-37 

21491 

3-93 

i  .  36864 

4.49 

1.50185 

5.05 

1.61939 

3.38 

21788 

3-94 

1.37118 

4.50 

^.  50408 

5.06 

1.62137 

3-39 

22083 

3.95 

I-3737I 

4-51 

i  .  50630 

5.07 

1.62334 

3-40 

22378 

3.96 

1.37624 

4.52 

i  .50851 

5.08 

1.62531 

3-4i 
3-42 

22671 
22964 

IS 

1.37877 
1.38128 

4.53 
4-54 

1.51072 
1-51293 

5.09 
S.io 

1.62728 
1.62924 

3-43 

23256 

3.99 

i-38379 

4-55 

I-5I5I3 

S.ii 

i  .63120 

3-44 

23547 

4.00 

1.38629 

4.56 

I-5I732 

5.12 

1-63315 

3-45 
3.46 

23837 
.24127 

4.01 

4.02 

1.38879 
1.39128 

4-58 

i-5i95i 
1.52170 

5-13 
5.14 

1-63511 
1-63705 

3-47 

.24415 

4.03 

1-39377 

4-59 

1.52388 

S.iS 

i  .  63900 

3.48 

•  247°3 

4.04 

1.39624 

4.60 

1.52606 

5-i6 

i  .  64094 

3-49 

.  24990 

4.05 

1.39872 

4.61 

1-52823 

5.i7 

1.64287 

3.50 

.25276 

4.06 

i  .40118 

4.62 

I-53039 

5.18 

i  .  64481 

3.5i 
3-52 

•25562 
.25846 

4.08 

i  .  40364 
i  .40610 

4.63 
4.64 

1-53256 
i-5347i 

5-19 
5.20 

i  .  64673 
i  .  64866 

3-53 

.26130 

4.09 

i  .  40854 

4.65 

i.53687 

5.21 

i  .  65058 

3-54 

.26412 

4.10 

1.41099 

4.66 

1.53902 

5-22 

1.65250 

3-55 

.26695 

4.11 

1-41342 

4.67 

1.54116 

5.23 

1.65441 

3.56 

.26976 

4.12 

1.41585 

4.68 

i  •  5433° 

5-24 

1.65632 

3-57 

.27257 

4.13 

1.41828 

4.69 

1-54543 

5.25 

1.65823 

3.58 

s 

•27536 

4.14 

1.42070 

4.70 

I-54756 

5-26 

1.66013 

3-59 

\j    \j^ 

.27815 

4.15 

1.42311 

4.71 

1.54969 

5-27 

i  .  66203 

3.6o 

.  28093 

4.16 

1-42552 

4.72 

i.55i8i 

5.28 

i  .  66393 

3-6i 

.28371 

4.17 

1.42792 

4-73 

1-55393 

5-29 

1.66582 

3.62 

.  28647 

4.18 

1-43031 

4-74 

1.55604 

5-30 

1.66771 

3.63 

.28923 

4.19 

1.43270 

4-75 

i-558i4 

5-31 

1.66959 

3-64 

.29198 

4.20 

1.43508 

4.76 

1.56025 

5.32 

1.67147 

3-65 

•  29473 

4.21 

I-43746 

4-77 

1-56235 

5-33 

I-67335 

3-66 

.29746 

4.22 

i  .  43984 

4.78 

1.56444 

5-34 

1.67523 

3-67 

.30019 

4.23 

1.44220 

4-79 

1-56653 

5-35 

1.67710 

3-68 

.30291 

4.24 

1.44456 

4.80 

1.56862 

5.36 

i  .  67896 

3.69 
3.70 

•30563 
•30833 

4.25 
4.26 

i  .  44692 
1.44927 

4.81 
4.82 

1.57070 
1.57277 

5-37 
5.38 

1.68083 
1.68269 

3-7i 

•3II03 

4.27 

1.45161 

4-83 

I-57485 

5-39 

1.68455 

3.72 

•3!372 

4.28 

1-45395 

4.84 

1.57691 

5-40 

i  .  68640 

3-73 

.31641 

4.29 

1.45629 

4-85 

1-57898 

5.4i 

i  .  68825 

3-74 

.31909 

4.30 

1.45861 

4.86 

1.58104 

5.42 

i  .69010 

3-75 

.32176 

4.31 

i  .  46094 

4.87 

i  .  58309 

5-43 

1.69194 

3-76 

.32442 

4.32 

1.46326 

4.88 

1-58515 

5-44 

1.69378 

3-77 

.32707 

4.33 

I-46557 

4.89 

1.58719 

5.45 

1.69562 

3-78 

.32972 

4.34 

1.46787 

4.90 

1.58924 

5.46 

1.69745 

3-79 

•33237 

1  4'3I 

i  .47018 

4.91 

1-59127 

5-47 

1.69928 

3.8o 

•3SSoo 

1  4.36 

1.47247 

4-92 

i-5933i 

5.48 

i  .  70111 

93 


TABLE  XTTT    Continued.  —  HYPERBOLIC    LOGARITHMS. 


N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

5-49 

1.70293 

6.05 

.  80006 

6.61 

.88858 

7.17 

.96991 

5.50 

1  •  7°475 

6.06 

.80171 

6.62 

.89010 

7.18 

•97I3° 

5-51 

.70656 

6.07 

•80336 

6.63 

.89160 

7.19 

.97269 

5-52 

.70838 

6.08 

.80500 

6.64 

.89311 

7.20 

.97408 

5-53 

.71019 

6.0Q 

.80665 

6.65 

.89462 

7.21 

•97547 

5-54 

.71199 

6.10 

.80829 

6.66 

.89612 

7.22 

.97685 

5-55 

.71380 

6.ii 

.  80993 

6.67 

.89762 

7-23 

.97824 

5.56 

.71560 

6.12 

.81156 

6.68 

.89912 

7.24 

.97962 

5-57 

.71740 

6.13 

.81319 

6.69 

.90061 

7-25 

.98100 

5.58 

.71919 

6.14 

.81482 

6.70 

.90211 

7.26 

.98238 

5-59 

.  72098 

6.15 

•81645 

6.71 

.90360 

7.27 

.98376 

5.6o 

.72277 

6.16 

.81808 

6.72 

•  90509 

7.28 

•98513 

5.6i 

•72455 

6.17 

.81970 

6.73 

.90658 

7.29 

.98650 

5-62 

*       S 

.72633 

6.18 

.82132 

6.74 

.90806 

7.30 

.98787 

5.63 

.72811 

6.19 

.82294 

6.75 

.90954 

7.31 

.98924 

5-64 

.  72988 

6.20 

•82455 

6.76 

.91102 

7-32 

.99061 

5.65 

.73166 

6.21 

.82616 

6.77 

.91250 

7-33 

.99198 

5-66 

'  73342 

6.22 

.82777 

6.78 

.91398 

7-34 

•99334 

5.67 

•735*9 

6.23 

•82937 

6.79 

•91545 

7-35 

•99470 

5.68 

•  73695 

6.24 

.83098 

6.80 

.91692 

7.36 

.99606 

5.69 

•73871 

6.25 

•83258 

6.81 

.91839 

7-37 

.99742 

5.70 

i  .  74047 

6.26 

.83418 

6.82 

.91986 

7.38 

•99877 

5-71 

1.74222 

6.27 

•83578 

6.83 

.92132 

7-39 

.00013 

5-72 

1  •  74397 

6.28 

•83737 

6.84 

.92279 

7.40 

2.00148 

5-73 

J-74572 

6.29 

.  83896 

6.85 

.92425 

7.41 

2  .  00283 

5-74 

.74746 

6.30 

•84055 

6.86 

.92571 

7.42 

2.O04I8 

5-75 

.  74920 

6.31 

.84214 

6.87 

.92716 

7-43 

2.00553 

5-76 

-  75°94 

6.32 

.84372 

6.88 

.92862 

7-44 

2.00687 

5-77 

.75267 

6.33 

•84530 

6.89 

.93007 

7.45 

2.00821 

5.78 

•75440 

6-34 

.84688 

6.90 

•93I52 

7-46 

2.00956 

5-79 

•75613 

6-35 

.84845 

6.91 

•93297 

7-47 

2  .01089 

5.8o 

.75786 

6.36 

.85003 

6.92 

•93442 

7.48 

2.01223 

5-8i 

•75958 

6-37 

.85160 

6-93 

•93586 

7-49 

2.01357 

5-82 

.76130 

6.38 

•85317 

6.94 

•93730 

7.50 

2.01400 

5-83 

•  76302 

6.39 

•85473 

6-95 

•93874 

7.5i 

2.01624 

5-84 

•  76473 

6.40 

•85630 

6.96 

.94018 

7.52 

2.01757 

5-85 

.  76644 

6.41 

.85786 

6-97 

.94162 

7-53 

2.01890 

5-86 

.76815 

6.42 

.85942 

6.98 

•94305 

7-54 

2  .O2O22 

•  76985 

6-43 

.86097 

6.99 

•  94448 

7-55 

2.02I55 

5  88 

> 

•77!56 

6.44 

•86253 

7.00 

•94591 

7.56 

2.02287 

5-89 

.77326 

6-45 

.  86408 

7.01 

•94734 

7-57 

2.02419 

5-90 

•77495 

6.46 

•86563 

7.02 

.94876 

7.58 

2.02551 

5-9i 

.77665 

6.47 

.86718 

7-03 

.95019 

7-59 

2.02683 

5-92 

•77834 

6.48 

.86872 

7.04 

.  -95l6i 

7.60 

2.02815 

5-93 

.  78002 

6-49 

.87026 

7.05 

•95303 

7.61 

2.02946 

5-94 

.78171 

6.50 

.87180 

7.06 

•95444 

7.62 

2.03078 

5-95 

.78339 

6.51 

•87334 

7-07 

.95586 

7.63 

2  .03209 

5.96 

.78507 

6.52 

.87487 

7.08 

•95727 

7.64 

2.03340 

5-97 

•78675 

6-53 

.87641 

7.09 

.95869 

7.65 

2.03471 

5.98 

.78842 

6-54 

87794 

7.10 

.96009 

7.66 

2.03601 

5-99 

.  79009 

6.55 

•87947 

7.11 

.96150 

7.67 

2.03732 

6.00 

.79176 

6.56 

.  88099 

7.12 

.96291 

7.68 

2.03862 

6.01 

•  79342 

6.57 

.88251 

7-13 

•96431 

7.69 

2.03992 

6.02 

•  795°9 

6.58 

88403 

7.14 

•96571 

7.70 

2.04122 

6.03 

•  79675 

6.59 

88555 

7-15 

.96711 

7.71 

2.04252 

6.04 

.  79840 

6.60 

88707 

7.16 

i  .06851 

7.72 

2.04.^1 

94 


TABLE  XIII.  Continued.  —  HYPERBOLIC    LOGARITHMS. 


N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

7-73 

2.04511 

8.30 

2.  11626 

8.87 

2.18267 

9.44 

2.24496 

7-74 

2.04640 

8.3I 

2.11746 

8.88 

2.18380 

9-45 

2  .24601 

7-75 

2.04769 

8.32 

2.II866 

8.89 

2.18493 

9.46 

2.24707 

7.76 

2.04898 

8.33 

2.II986 

8.90 

2.18605 

9-47 

2.24^13 

7-77 

2.05027 

8.34 

2.  I2I06  •« 

8.91 

2.18717 

9.48 

2.24918 

7.78 

2.05156 

8.35 

2.  12226 

8.92 

2.18830 

9.49 

2.25024 

7-79 

2.05284 

8.36 

*..I2346 

^•93 

2.  18942 

9-50 

2.25129 

7.80 

2.05412 

8.37 

2.  12465 

8.94 

2.19054 

9-51 

2.25234 

7.81 

2.05540 

8.38 

2.12585 

8-95 

2.19165 

9-52 

2.25339 

7.82 

2.05668 

8.39 

2.12704 

8.96 

2.19277 

9-53 

2.25444 

7.83 

2.05796 

8.40 

2^.  12823 

8-97 

2.19389 

9-54 

2.25549 

7.84 

2.05924 

8.41 

2.  12942 

8.98 

2  .  19500 

9-55 

2.25654 

7.85 

2.06051 

8.42 

2.  13061 

8.99 

2  .  19611 

9.56 

2.25759 

7.86 

2.06179 

8-43 

2.13180 

9.00 

2.19722 

9.57 

2  •  25863 

7.87 

2.06306 

8.44 

2.13298 

9.01 

2.19834 

9.58 

2.25968 

7.88 

2.06433 

8.45 

2.13417 

9.02 

2.19944 

9-59 

2.26072 

7.89 

2.06560 

8.46 

2-13535 

9-03 

2.20055 

9.60 

2.26176 

7.90 

2.06686 

8.47 

2.13653 

9.04 

2.20166    l|     9.6l 

2.26280 

7.91 

2.06813 

8.48 

2.13771 

9.05 

2  .20276 

9.62 

2.26384 

7.92 

2  .  06939 

8-49 

2.13889 

9.06 

2.20387 

9-63 

2.26488 

7-93 

2.07065 

8.50 

2.14007 

9.07 

2.20497 

9.64 

2.26592 

7-94 

2.07191 

8.51 

2.I4I24 

9.08 

2.2O607 

9.65 

2.26696 

7-95 

2.07317 

8.52 

2.14242 

9.09 

2.20717 

9.66 

2.26799 

7.96 

2.07443 

8.53 

2.14359 

9.10 

2.2O827 

9.67 

2  .  26903 

7*98 

2.07568 
2.07694 

8.54 
8.55 

2.14476 
2.14593 

9.11 
9.12 

2.20937 
2  .2IO47 

9.68 
9.69 

2  .27006 
2.27109 

7-99 

2.07819 

8.56 

2.  I47IO 

9.13 

2.2II57 

9.70 

2     27213 

8.00 

2.07944 

8.57 

2.14827 

9.14 

2  .21266 

9.71 

2.27316 

8.01 

2  .08069 

8.58 

2  .  14943 

9-15 

2-21375 

9.72 

2.27419 

8.02 

2.08194 

8.59 

2.  15060 

9.16 

2.21485 

9-73 

2.27521 

8.03 

2.08318 

8.60 

2.15176 

9.17 

2.21594 

9.74 

2.27624 

8.04 

2.08443 

8.61 

2.15292 

9.18 

2.21703 

9-75 

2.27727 

8.05 

2.08567 

8.62 

2.15409 

9.19 

2  .2l8l2 

9.76 

2.27829 

8.06 

2.08691 

8.63 

2.I5524 

9.20 

2  .21920 

9.77 

2.27932 

8.07 

2.08815 

8.64 

2.  15640 

9.21 

2  .  22029 

9.78 

2.28034 

8.08 

2.08939 

8.65 

2-I5756 

9.22 

2.22138 

9-79 

2.28136 

8.09 

2  .  09063 

8.66 

2.15871 

9-23 

2  .22246 

9.80 

2.28238 

8.10 

2.09186 

8.67 

2.15987 

9.24 

2.22351 

9.81 

2.28340 

8.ii 

2.09310 

8.68 

2.  l6l02 

9-25 

2.22462 

9.82 

2  .28442 

8.12 

2  •  09433 

8.69 

2.I62I7 

9.26 

2.  22570 

9-83 

2.28544 

8.13 

2.09556 

8.70 

2.16332 

9.27 

2.22678 

9.84 

2.28646 

8.14 

2.09679 

8.71 

2.16447 

9.28 

2.22786 

9.85 

2.28747 

8.15 

2.09802 

8.72 

2.  16562 

9.29 

2.22894 

9.86 

2.28849 

8.16 

2.09924 

8.73 

2.  16677 

9-30 

2.  23OOI 

9.87 

2.28950 

8.17 

2  .  10047 

8.74 

2.  16791 

9-31 

2.23109 

9.88 

2.29051 

8.18 

2.  10169 

8-75 

2.16905 

9-32 

2.23216 

9.89 

2.29152 

8.19 

2.IO29I 

8.76 

2  .  I702O 

9-33 

2.23323 

9.90 

2.29253 

8.20 

2.I04I3 

8.77 

2.17134 

9-34 

2.23431 

9.91 

2.29354 

8.21 

2-10535 

8.78 

2  .  17248 

9-35 

2.23538 

9.92 

2.29455 

8.22 

2.  10657 

8.79 

2.17361 

9.36 

2.23645 

9-93 

2.29556 

8.23 

2.10779 

8.80 

2.17475 

9-37 

2.23751 

9.94 

2-29657 

8.24 

2.10900 

8.81 

2.17589 

9-38 

2.23858 

9-95 

2.29757 

8.25 

2.  IIO2I 

8.82 

2.17702 

9-39 

2.23965 

9.96 

2.29858 

8.26 

2.III42 

8.83 

2.17816 

9.40 

2.24071 

9-97 

2.29958 

8.27 

2.II263 

8,84 

2.17929 

9.41 

2.24177 

9.98 

2.30058 

8.28 

2.  11384 

8.85 

2.  18042 

9.42 

2  .  24184 

9.99 

2.30158 

8.29 

2.11=^05 

8.86 

2.l8l55 

9-43 

2  .  24.390 

95 


LOGARITHMS   O7  ITUMBERS. 


No. 

O 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

100 

OO  000 

043 

087 

I30 

173 

217 

260 

3°3 

346 

389 

101 

432 

475 

518 

604 

647 

689 

732 

775 

817 

102 

860 

9°3 

945 

988 

*o3o 

*072 

*ii5 

*I57 

*i99 

*242 

44 

43 

42 

103 

oi  284 

326 

368 

410 

452 

494 

536 

578 

620 

662 

i 

88 

4-3 

8/C 

si 

104 

7°3 

745 

787 

828 

870 

912 

953 

995 

*o36 

=1=078 

2 

3 

O.  O 

13.2 

.  O 

12.9 

0.4 

12.6 

105 

02  119 

1  60 

202 

243 

284 

325 

366 

407 

449 

490 

4 

17.6 

17.2 

16.8 

106 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

^ 

22.0 
26.4 

21.5 
25.8 

21.0 
25.  2 

107 

938 

979 

*oi9 

*o6o 

*IOO 

*i8i 

*222 

*262 

*302 

7 

30.8 

30.1 

29.4 

1  08 

03  342 

383 

423 

463 

5°3 

543 

583 

623 

663 

7°3 

8 
9 

35-2 
39-6 

34-4 
38.7 

33-6 
37.8 

109 

743 

782 

822 

862 

902 

941 

981 

*02I 

*o6o 

^100 

no 

04  139 

179 

218 

258 

297 

336 

376 

415 

454 

493 

in 

S32 

571 

610 

650 

689 

727 

766 

844 

883 

112 

922 

961 

999 

*o38 

*077 

*ii5 

*i54 

*I92 

*23I 

*269 

41 

40 

39 

113 

05  308 

346 

385 

423 

461 

500 

538 

576 

614 

652 

2 

tl 

4*  o 
8.0 

3-  9 

7.8 

114 

690 

729 

767 

805 

843 

88  1 

918 

956 

994 

*032 

3 

12.3 

12.0 

11.7 

115 

06  070 

108 

145 

183 

221 

258 

296 

333 

371 

408 

4 
5 

16.4 

20.5 

16.0 
20.  o 

iS-6 
iQ-5 

116 

446 

483 

521 

558 

595 

*633 

670 

707 

744 

78l 

6 

24.6 

24.0 

23-4 

117 

819 

856 

893 

93° 

967 

*04i 

$078 

*n5 

*I5I 

g7 

28.7 
32.8 

28.0 
32.0 

27-3 

31  •  2 

118 

07  188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

9 

o*  *-* 

36.9 

36.0 

35-1 

119 

555 

591 

628 

664 

700 

*737 

773 

809 

846 

882 

120 

918 

954 

990 

*027 

*i35 

*i7i 

*207 

*243 

121 

08  279 

314 

35° 

386 

422 

458 

493 

529 

565 

600 

-O 

tft 

122 

636 

672 

707 

743 

778 

814 

849 

884 

920 

955 

i 

fa 

37 

3-7 

3° 

3-6 

123 

991 

*026 

*o6i 

*o96 

*I32 

*i67 

*202 

*237 

*272 

*3°7 

2 

7-6 

7-4 

7-2 

124 

09  342 

377 

412 

447 

482 

5*7 

552 

587 

621 

656 

3 

4 

11.4 

ii.  i 

14.8 

10.8 
14.4 

125 

691 

726 

760 

795 

830 

864 

899 

934 

968 

*oo3 

5 

19.0 

18.5 

18.0 

126 

10  037 

072 

106 

140 

175 

209 

243 

278 

3I2 

346 

6 

7 

22.8 
26.6 

22.2 
25.9 

21.6 
2<5.  2 

127 

380 

4I5 

449 

483 

5Z7 

55i 

585 

619 

653 

*68? 

8 

30-4 

29.6 

28.8 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992 

9 

34-2 

33-3 

32.4 

129 

ii  059 

093 

126 

160 

193 

227 

26l 

294 

327 

III 

130 

394 

428 

461 

494 

528 

561 

594 

628 

66  1 

694 

727 

760 

793 

826 

860 

893 

926 

959 

992 

*024 

35 

34 

33 

132 

12  057 

090 

123 

156 

189 

222 

254 

287 

320 

352 

I 

3-5 

3-4 

3-3 

133 

385 

418 

45° 

483 

5l6 

548 

613 

646 

678 

2 

3 

7.0 

10.  5 

6.8 

IO.  2 

6.6 
9-  9 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969 

*OOI 

4 

14.0 

13-6 

13.2 

135 

13  °33 

066 

098 

130 

162 

194 

226 

258 

290 

322 

i 

21  .0 

17.0 
2O-4 

16.5 
19.8 

136 

354 

3*86 

418 

45° 

481 

513 

545 

577 

609 

640 

7 

24-5 

23-8 

23.1 

138 

672 
988 

704 

735 
*o5i 

767 

*082 

799 
*ii4 

830 

862 
*i76 

893 

*2o8 

925 

956 

*270 

8 
9 

28.0 
31.5 

27.2 
30.6 

26.4 
29.7 

139 

14  301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

140 

613 

644 

675 

706 

737 

*768 

799 

829 

860 

89I 

141 

922 

953 

983 

*oi4 

*°45 

*io6 

*i37 

*i68 

*i98 

32 

31 

30 

142 

15  229 

259 

290 

320 

351 

381 

412 

442 

473 

5°3 

I 

2 

3-2 
6.4 

6^2 

3-0 
6.0 

143 

534 

564 

594 

625 

655 

685 

7J5 

746 

776 

806 

3 

9.6 

9-3 

9.0 

144 

836 

866 

897 

927 

957 

987 

*oi7 

*077 

*I07 

4 

12.8 

12.4 

12.0 

145 

16  137 

167 

197 

227 

256 

286 

316 

346 

376 

406 

I 

;i6.  o 
19.2 

iS-S 
18.6 

ll'.O 

I46 

435 

465 

495 

52^ 

554 

584 

613 

643 

673 

702 

I 

22.^ 

21.7 

1.   V 

21.0 

147 

732 

761 

791 

820 

85c 

879 

909 

938 

967 

997 

o 
9 

25.6 
28.8 

24-  c 

27.  g 

24.0 
27.0 

I48 

17  026 

056 

085 

114 

143 

173 

202 

231 

260 

289 

149 

3*9 

348 

377 

406 

435 

464 

493 

522 

55i 

S8o 

LOGARITHMS  OF  NUMBERS. 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

ISO 

17  609 

638 

667 

696 

725 

754 

*fo 

8n 

840 

869 

898 

926 

955 

984 

*099 

*I27 

*i56 

152 

1  8  184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

29 

28 

153 

469 

498 

526 

554 

583 

639 

667 

696 

724 

i 

2 

2.9 

r  8 

2.8 

154 

752 

780 

808 

837 

865 

893 

921 

949 

977 

*°°5 

3 

5-° 

8.7 

5  •  o 

8.4 

155 

19  033 

061 

089 

ii  7,, 

145 

1734' 

201 

229 

257 

285 

4 

ii.  6 

II  .  2 

156 

312 

340 

368 

39  6 

424 

4^i 

479 

5°7 

535 

562 

I 

14-5 
17-4 

14.0 

16.8 

157 

590 

618 

645 

673 

700 

*7*8 

*o5o 

783 

8n 

838 

I 

20.3 

19.  6 

158 

866 

893 

921 

948 

976 

*o58 

*o85 

*II2 

8 
9 

23.2 
26.1 

22.4 
25.2 

159 

20  140 

167 

194 

222 

249 

276 

3°3 

33° 

358 

385 

1  60 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

161 

683 

710 

737 

763 

79° 

817 

844 

871 

898 

925 

162 

952 

978 

*oos 

*032 

*°59 

*o85 

*II2 

*i39 

*i65 

*I92 

27 

26 

2.  6 

163 

21  219 

245 

272 

299 

325 

352 

378 

43  1 

458 

2 

2  .  7 

5-4 

5-2 

164 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 

3 

8.  i 

7-8 

165 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

4 

10.8 
13-5 

10.4 
13-0 

1  66 

22  Oil 

°37 

063 

089 

IT5 

141 

l67 

194 

220 

246 

6 

16.2 

15.6 

167 

272 

298 

324 

35° 

376 

401 

427 

453 

479 

505 

I 

18.9 

21.6 

18.2 

20.8 

1  68 

531 

557 

583 

608 

634 

660 

686 

712 

737 

*763 

9 

24-3 

23-4 

169 

789 

814 

840 

866 

891 

917 

943 

963 

994 

170 

23  045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

171 

300 

325 

35° 

376 

401 

426 

452 

477 

502 

528 

•)  £ 

172 

553 

578 

603 

629 

654 

679 

704 

729 

*754 

779 

25 

I    2.5 

173 

805 

830 

855 

880 

9°5 

93° 

955 

980 

+030 

2  5.0 

174 

24  055 

080 

I30 

155 

180 

204 

229 

254 

279 

3  7-5 
4  10.  o 

175 

3°4 

329 

353 

378 

403 

428 

452 

477 

502 

527 

512.5 

176 

551 

576 

601 

625 

650 

674 

699 

724 

748 

773 

O  15.0 

7  17.5 

177 

797 

822 

846 

87I 

895 

920 

944 

969 

993 

*oi8 

8  2O.O 

178 

25  042 

066 

091 

"5 

139 

164 

1  88 

212 

237 

261 

9  22.5 

179 

285 

310 

334 

358 

382 

406 

43  1 

455 

479 

5°3 

180 

•  527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

181 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

24 

23 

182 

26  007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

I 

2.4 

2.3 

183 

245 

269 

293 

3l6 

340 

364 

387 

411 

435 

458 

2 

3 

4.8 
7  ^ 

4.6 
6.  9 

184 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

4 

9.6 

9-2 

185 

717 

74i 

764 

788 

811 

834 

858 

881 

9°5 

928 

I 

12.0 

ii.  5 

1  86 

975 

998 

*02I 

*°45 

*o68 

*o9i 

*ii4 

*I38 

*i6i 

o 

7 

14.4 

16.8 

13.8 
16.1 

187 

27  184 

207 

231 

254 

277 

300 

323 

346 

37° 

393 

8 

19.2 

18.4 

1  88 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

9 

21  .  6 

20.7 

189 

646 

669 

692 

715 

738 

761 

784 

*8°7 

830 

852 

190 

875 

898 

921 

944 

967 

989 

*OI2 

*o58 

*o8i 

191 

28  103 

126 

149 

171 

194 

217 

240 

262 

285 

3°7 

22 

21 

192 

33° 

353 

375 

398 

42! 

443 

466 

488 

511 

533 

x 

2 

2.2 

2.  I 

193 

556 

578 

60! 

623 

646 

668 

69I 

7*3 

735 

758 

3 

4  •  A 
6.6 

4*  2 
6-3 

194 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

4 

8.8 

8.4 

195 

29  003 

026 

048 

070 

092 

"5 

137 

159 

181 

203 

I 

;I  I  .  O 
13-2 

10.  5 

12.6 

196 

226 

248 

270 

292 

336 

358 

380 

403 

425 

I 

15-4 

xi'o 

197 

447 

469 

491 

535 

557 

579 

601 

623 

645 

8 
9 

17.6 
19.8 

16.8 
18.9 

198 

667 

688 

710 

732 

754 

776 

798 

820 

842 

863 

199 

885 

907 

929 

951 

973 

994 

*oi6 

#038 

*o6o 

*o8i 

3<> 

97 


-LOGARITHMS  OF  NUMBERS. 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

200 

30  103 

125 

146 

1  68 

190 

211 

233 

255 

276 

298 

201 

320 

34i 

363 

384 

406 

428 

449 

471 

492 

5i4 

202 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

22   21 

203 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

I   2.2   2.1 

2   4442 

204 

963 

984 

*oo6 

*027 

*o48 

*o69 

*09i 

*II2 

*i33 

*i54 

3  6.6  6.3 

205 

3i  i75 

197 

218 

239 

260 

281 

302 

323 

345 

366 

4  8.8  8.4 

206 

387 

408 

429 

45° 

471 

492 

513 

534 

555 

576 

5  1  1  .  o  10.5 
6  13.2  12.6 

207 

597 

618 

639 

660 

681 

702 

'723 

744 

.765 

785 

7  iS-4  i4-7 

208 

806 

827 

848 

869 

890 

911 

93i 

952 

973 

994 

8  17.6  16.  8 
9  19.8  18.9 

20Q 

32  OI5 

°35 

056 

077 

098 

118 

139 

1  60 

181 

201 

210 

222 

243 

263 

284 

3°5 

325 

346 

366 

387 

408 

211 

428 

449 

469 

49° 

5i° 

53i 

552 

572 

593 

6l3 

212 

634 

654 

675 

695 

7J5 

736 

756 

777 

797 

818 

20 

213 

838 

858 

879 

899 

919 

940 

960 

980 

*OOI 

*02I 

2 

4.0 

214 

33  <HI 

062 

082 

102 

122 

143 

163 

183 

203 

224 

3 

6.0 

215 

244 

264 

284 

3°4 

325 

345 

365 

385 

405 

425 

4 

8.  o 

10.  0 

216 

445 

465 

486 

506 

526 

546 

566 

586 

606 

626 

6 

12  .  O 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

I 

14.0 

16.0 

218 

846 

866 

885 

905 

925 

945 

965 

985 

*oo5 

*025 

9 

18.0 

219 

34  044 

064 

084 

104 

124 

*43 

163 

183 

203 

223 

220 

242 

262 

282 

301 

32I 

34i 

361 

380 

400 

420 

221 

439 

459 

479 

498 

518 

537 

557 

577 

596 

616 

TO 

222 

635 

655 

674 

694 

7J3 

733 

753 

772 

792 

8u 

i 

iy 

1.9 

223 

830 

850 

869 

889 

908 

928 

947 

967 

986 

*oo5 

2 

3-8 

224 

35  025 

044 

064 

083 

102 

122 

141 

1  60 

1  80 

199 

3 

4 

5-7 
7.6 

225 

218 

238 

257 

276 

295 

3J5 

334 

353 

372 

392 

5 

9-5 

226 

411 

43° 

449 

468 

488 

5°7 

526 

545 

564 

583 

6 

7 

11.4 
13-3 

227 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

8 

15-2 

228 

793 

813 

832 

851 

870 

889 

908 

927 

946 

965 

9 

17.1 

229 

984 

*oo3 

*02I 

*040 

*o59 

*078 

*o97 

*n6 

*i35 

*i54 

230 

36  173 

192 

211 

229 

248 

267 

286 

3°5 

324 

342 

231 

361 

380 

399 

418 

436 

455 

474 

493 

5" 

530 

18 

232 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

i 

1.8 

233 

736 

754 

773 

791 

810 

829 

847 

866 

884 

9°3 

2 

3 

3-6 

5-4 

234 

922 

940 

959 

977 

996 

*oi4 

*o33 

*°5i 

*o7o 

*oS8 

4 

7-  2 

235 

37  107 

125 

144 

162 

181 

199 

218 

236 

254 

273 

65 

9.0 
10.8 

236 

291 

31° 

328 

346 

365 

383 

401 

420 

438 

457 

7 

12.6 

237 

475 

493 

Sii 

53° 

548 

566 

585 

603 

621 

639 

8 

14.4 
16  2 

238 

658 

676 

694 

712 

73i 

749 

767 

785 

803 

822 

9 

239 

840 

858 

876 

894 

912 

931 

949 

967 

985 

*oo3 

240 

38  02  1 

039 

°57 

075 

093 

112 

130 

148 

1  66 

184 

241 

202 

220 

238 

256 

274 

292 

310 

328 

346 

364 

17 

242 

782 

399 

417 

435 

453 

471 

489 

507 

525 

543 

i 

2 

i-7 
3-  4 

243 

56l 

578 

596 

614 

632 

650 

668 

686 

703 

721 

3 

5-i 

244 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

4 

6.8 
c  - 

245 

917 

934 

952 

970 

987 

*oo5 

*023 

*04i 

*o58 

*076 

i 

«•  5 
IO.  2 

246 

39  °94 

in 

129 

146 

164 

182 

199 

217 

235 

2^2 

7 

ii.  9 

247 

270 

287 

3°5 

322 

340 

358 

375 

393 

410 

428 

9 

13.0 
15-3 

248 

445 

463 

480 

498 

5i5 

533 

55° 

568 

585 

602 

249 

620 

637 

655 

672 

690 

707 

724 

742 

759 

777 

98 


LOGARITHMS   OF   NUMBERS. 


»0. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  PU. 

250 

39  794 

811 

829 

846 

863 

881 

898 

9i5 

933 

95° 

251 

967 

985 

*002 

*OIQ 

*o37 

*°54 

*o7i 

*o88 

*io6 

252 

40  140 

157 

175 

192 

209 

226 

243 

261 

278 

295 

18 

253 

312 

329 

346 

364 

381. 

398 

4i5 

432 

449 

466 

i 

1.8 

_  -; 

254 

483 

500 

518 

535 

552 

5.69 

586 

603 

620 

637 

2 

3 

3-° 

5-4 

255 

654 

671 

688 

IPS 

722 

756 

773 

79° 

807 

4 

7.2 

256 

824 

841 

858 

875 

892 

9°9 

926 

943 

960 

976 

j 

9-0 
10.8 

257 

993 

*OIO 

*O27 

*o44 

*o6i 

*o78 

*°95 

*in 

*I28 

*i45 

7 

12.6 

258 

41  162 

179 

196 

212 

229 

246 

263 

280 

296 

8 

Q 

14.4 
16.2 

259 

33° 

347 

363 

380 

397 

414 

43° 

447 

464 

481 

260 

497 

531 

547 

564 

581 

597 

614 

631 

647 

261 

664 

68  1 

697 

714 

747 

764 

780 

797 

814 

262 

830 

847 

863 

880 

896 

913 

929 

946 

963 

979 

17 

263 

996 

*OI2 

*02Q 

*045 

*062 

$078 

*in 

*I27 

*i44 

I 
2 

1-7 
3-  4 

264 

42  1  60 

177 

193 

210 

226 

243 

259 

275 

292 

308 

3 

265 

325 

341 

357 

374 

39° 

406 

423 

439 

455 

472 

4 

• 

6^8 
8-5 

266 

488 

5°4 

521 

537 

553 

57° 

586 

602 

619 

635 

6 

10.2 

267 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

7 

ii.  9 

TO   f\ 

268 

813 

830 

846 

862 

878 

894 

911 

927 

*943 

959 

9 

13.  o 

15-3 

269 

975 

991 

*oo8 

*024 

*04o 

^056 

*o88 

*I20 

270 

43  *36 

169 

185 

201 

217 

233 

249 

265 

28l 

271 

297 

3J3 

329 

345 

361 

377 

393 

409 

425 

441 

*JC 

272 

457 

473 

489 

5°5 

521 

537 

553 

569 

584 

600 

ID 

1.6 

273 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

2 

3-2 

274 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

3 

4.8 
6  4 

275 

933 

949 

965 

981 

996 

*OI2 

*028 

*o44 

*°59 

5 

S'.Q 

276 

44  091 

107 

122 

138 

154 

I70 

185 

201 

217 

232 

6 

9.6 

277 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

I 

ll'.l 

278 

404 

420 

436 

45  1 

467 

483 

498 

514 

529 

545 

9 

14.4 

279 

560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 

281 

871 

886 

902 

917 

932 

948 

963 

979 

994 

*OIO 

282 

45  025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

I 

i-S 

283 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

2 

3-o 

284 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

4 

4-  5 
6.0 

285 

484 

500 

515 

53° 

545 

561 

576 

591 

606 

621 

3 

7-5 

286 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

0 
7 

9.0 
10.5 

287 

788 

803 

818 

834 

849 

864 

879 

*894 

909 

924 

8 

12.0 

288 

939 

954 

969 

984 

*000 

*OI5 

*030 

*o6o 

9 

13-5 

289 

46  090 

105 

120 

135 

150 

165 

180 

195 

210 

225 

290 

240 

255 

270 

285 

300 

315 

33° 

345 

359 

374 

291 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

14 

292 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672 

I 

2 

1.4 

2.8 

293 

687 

702 

7l6 

73  1 

746 

761 

776 

79° 

805 

820 

3 

4-2 

294 

835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

4 

5-6 

295 

982 

997 

*OI2 

*026 

*o4i 

*o56 

*o7o 

#085 

*IOO 

*ii4 

1 

7-0 
8.4 

296 

47  129 

144 

159 

173 

1  88 

202 

217 

232 

246 

261 

z 

9.8 

297 

276 

290 

3°5 

3*9 

334 

349 

363 

378 

392 

407 

o 
9 

II  .  2 
12.6 

298 

422 

436 

451 

465 

480 

494 

5°9 

524 

538 

553 

299 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

LOGARITHMS   OF  NUMBERS. 


No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

300 

47  712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

301 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 

302 

48  ooi 

029 

044 

058 

°73 

087 

101 

116 

130 

303 

144 

J59 

173 

187 

202 

216 

230 

244 

259 

273 

304 

287 

302 

3l6 

330 

344 

359 

373 

387 

401 

416 

305 

430 

444 

458 

473 

487 

501 

515 

53° 

544 

558 

IS 

306 

572 

586 

601 

615 

629 

643 

657 

671 

686 

700 

i 

2 

3-° 

307 

714 

728 

742 

756 

77° 

785 

799 

813 

827 

841 

3 

4-5 

308 

855 

869 

883 

897 

911 

926 

940 

954 

968 

982 

4 

6.0 

7-5 

309 

996 

*OIO 

*024 

*o38 

1=052 

*o66 

*o8o 

*094 

*io8 

*I22 

6 

9.0 

310 

49  I36 

15° 

164 

178 

192 

206 

220 

234 

248 

262 

I 

10.5 

3" 

276 

290 

3°4 

3^8 

332 

346 

360 

374 

388 

4O2 

9 

13-5 

312 

415 

429 

443 

457 

471 

485 

499 

513 

527 

54i 

554 

568 

582 

596 

610 

624 

638 

651 

665 

679 

3J4 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 

315 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 

316 

969 

982 

996 

*OIO 

*024 

*o37 

*°79 

*092 

14 

3*7 

50  106 

120 

133 

147 

161 

174 

188 

202 

215 

229 

i 

2 

1.4 

2.8 

318 

243 

256 

270 

284 

297 

3" 

325 

338 

352 

365 

3 

4.2 

319 

379 

393 

406 

420 

433 

447 

461 

474 

488 

501 

4 

5-6 

320 

529 

542 

556 

569 

583 

596 

610 

623 

637 

1 

7.  o 
8.4 

321 

651 

664 

678 

691 

7°5 

718 

732 

745 

759 

772 

7 

9.8 

322 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

9 

II  .  2 
12.6 

323 

920 

934 

947 

961 

974 

987 

*OOI 

*oi4 

*028 

*o4i 

324 

5i  °55 

068 

081 

°95 

108 

121 

135 

148 

162 

175 

325 

188 

202 

215 

228 

242 

255 

268 

282 

295 

308 

326 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 

327 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 

13 

328 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

i 

2  6 

329 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

2 

3 

3-9 

330 

851 

865 

878 

891 

904 

917 

93° 

943 

957 

97° 

4 

5-2 

6M 

33i 

983 

996 

*oo9 

*O22 

*o6i 

*o75 

*oS8 

*IOI 

5 
6 

•  5 
7.8 

332 

52  114 

127 

140 

153 

1  66 

179 

192 

205 

218 

231 

I 

9.1 

333 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 

8 
9 

10.4 
II.  7 

334 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

335 

53° 

543 

556 

569 

582 

595 

608 

621 

336 

634 

647 

660 

673 

686 

699 

711 

724 

737 

75° 

337 

763 

776 

789 

802 

8i5 

827 

840 

853 

866 

879 

338 

892 

9°5 

917 

93° 

943 

956 

969 

982 

994 

*oo7 

12 

339 

53  °2o 

°33 

046 

058 

071 

084 

097 

no 

122 

135 

i 

1.2 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

2 

3 

2.  4 

3-6 

275 

288 

301 

314 

326 

339 

352 

364 

377 

39° 

4 

4.8 

342 

403 

4*5 

428 

441 

453 

466 

479 

4Q  i 

504 

517 

i 

6.0 

7-  2 

343 
344 

529 
656 

542 
668 

m 

567 
694 

706 

593 
719 

605 

732 

618 

744 

631 
757 

643 
769 

7 
8 

8.4 
9.6 
10.8 

345 

782 

794 

8o7 

820 

832 

845 

857 

870 

882 

895 

346 

908 

920 

933 

945 

958 

970 

983 

995 

*oo8 

*020 

347 

54  033 

045 

058 

070 

083 

°95 

1  08 

120 

133 

145 

348 

158 

170 

183 

195 

208 

220 

233 

245 

2S8 

270 

349 

283 

295 

3°7 

320 

332 

345 

357 

37° 

382 

394 

100 


LOGARITHMS  OK 


No. 

o 

I 

2 

3 

4' 

'  S'\ 

;ft  f 

;7C 

f  ft;  . 

''$  '' 

p» 

.  PU. 

350 

54  407 

419 

432 

444 

456 

469 

481 

494 

506 

5i8 

351 

53i 

543 

555 

568 

580 

593 

605 

617 

630 

642 

352 

654 

667 

679 

691 

704 

716 

728 

74i 

753 

765 

353 

777 

790 

802 

814 

827 

839 

851 

864 

876 

888 

354 

900 

9i3 

925 

937 

949V 

962 

974 

986 

998 

*on 

355 

55  023 

°35 

047 

060 

072 

§34 

096 

108 

121 

133 

13 

356 

145 

157 

169 

182 

194 

206 

218 

230 

242 

255 

i 

2 

5:1 

357 

267 

279 

291 

3°3 

315 

328 

340 

352 

364 

376 

3 

3-9 

358 

388 

400 

4i3 

425 

437 

449 

461 

473 

485 

497 

4 
5 

6*5 

359 

509 

522 

534 

546 

558 

570 

582 

594 

606 

618 

6 

7.8 

360 

630 

642 

654 

666 

678 

691 

7°3 

7i5 

727 

739 

I 

9.1 

36i 

75i 

763 

775 

787 

799 

811 

823 

835 

847 

859 

9 

10.  4 

11.7 

362 

871 

883 

895 

907 

919 

931 

943 

955 

967 

979 

363 

991 

*oo3 

*OI5 

*027 

*o38 

*c>5o 

*o62 

*074 

*o86 

*098 

364 

56  no 

122 

i34 

146 

158 

170 

182 

194 

205 

217 

365 

229 

241 

253 

265 

277 

289 

301 

312 

324 

336 

366 

348 

360 

372 

384 

396 

407 

419 

431 

443 

455 

12 

367 

467 

478 

49° 

502 

5*4 

526 

53s 

549 

56i 

573 

I 
2 

1.2 

2.4 

368 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

3 

3-6 

369 

7°3 

714 

726 

738 

75° 

761 

773 

785 

797 

808 

4 

C 

4-8 
6  o 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

6 

7.2 

37i 

937 

949 

961 

972 

984 

996 

*oo8 

*oi9 

*o3i 

*o43 

I 

8.4 
9/5 

372 

57  °54 

066 

078 

089 

101 

"3 

124 

136 

148 

159 

9 

.0 

10.8 

373 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 

374 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 

375 

403 

4i5 

426 

438 

449 

461 

473 

484 

496 

5°7 

376 

5*9 

53° 

542 

553 

565 

576 

588 

600 

611 

623 

377 

634 

646 

657 

669 

680 

692 

7°3 

7i5 

726 

738 

II 

378 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 

I 

i.i 

379 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

2 

3 

3-3 

380 

978 

990 

*OOI 

*oi3 

*O24 

*o35 

*o47 

*o58 

*o7o 

*o8i 

4 

4-4 

38i 

58  092 

104 

n5 

127 

138 

149 

161 

172 

184 

195 

i 

5-  5 
6.6 

382 

206 

218 

229 

240 

252 

263 

274 

286 

297 

3°9 

I 

11 

383 

320 

33i 

343 

354 

365 

377 

388 

399 

410 

422 

0 

9 

o.  o 
9-9 

384 

.   433 

444 

456 

467 

478 

490 

5°i 

512 

524 

535 

385 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 

386 

659 

670 

681 

692 

704 

7i5 

726 

737 

749 

760 

387 

782 

794 

805 

816 

827 

838 

850 

861 

872 

388 

883 

894 

906 

917 

928 

939 

95° 

961 

973 

984 

10 

389 

995 

*oo6 

*oi7 

*028 

*040 

*o5i 

*062 

*073 

*o84 

*o95 

i 

I  0 

390 

59  106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

2 

3 

2  O 

3  ° 

39i 

218 

229 

240 

251 

262 

273 

284 

295 

306 

3i8 

4 

4  o 

392 

329 

340 

35i 

362 

373 

384 

395 

406 

417 

428 

i 

S  ° 
6.0 

393 

439 

45° 

461 

472 

483 

494 

506 

5i7 

528 

539 

7 

7.0 

394 

55° 

56i 

572 

583 

594 

605 

616 

627 

638 

649 

8 

8.0 

9O 

395 

660 

671 

682 

693 

704 

715 

726 

737 

748 

759 

•  *"* 

396 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

397 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 

398 

988 

999 

*OIO 

*O2I 

*032 

*o43 

*054 

*o65 

4=076 

*o86 

399 

60  097 

108 

119 

I30 

141 

IS2 

163 

173 

184 

195 

XOI 


,  ^        LOGAKIOTHMS  OF  NUMBERS. 


No. 

\  Q  ^  ; 

ill 

2t  i  : 

Uj 

V4.  ' 

5 

6 

7 

8 

9 

Pp.  Pts. 

400 

60  206 

217 

228 

239 

249 

260 

271 

282 

293 

3°4 

401 

314 

325 

336 

347 

35S 

369 

379 

39° 

401 

412 

402 

423 

433 

444 

455 

466 

477 

487 

498 

5°9 

520 

403 

S3i 

54i 

552 

563 

574 

584 

595 

606 

617 

627 

404 

638 

649 

660 

670 

681 

692 

7°3 

7i3 

724 

735 

405 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 

406 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

407 

959 

970 

981 

991 

*002 

*OI3 

*023 

*034 

*o45 

*°55 

408 

61  066 

077 

087 

098 

I09 

119 

130 

140 

I51 

162 

i 

II 

i.i 

409 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

2 

2.2 

410 

278 

289 

300 

310 

32I 

33i 

342 

352 

363 

374 

3 

3-3 

411 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

5 

4.  4 

5.5 

4I2 

490 

500 

511 

52i 

532 

542 

553 

563 

574 

584 

6 

f 

6.6 

413 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

8 

7  •  7 
8.8 

414 

700 

711 

721 

73i 

742 

752 

763 

773 

784 

794 

9 

9-9 

415 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 

416 

909 

920 

93° 

941 

951 

962 

972 

982 

993 

*°°3 

417 

62  014 

024 

034 

°45 

055 

066 

076 

086 

097 

107 

4l8 

118 

128 

138 

149 

159 

170 

1  80 

190 

201 

211 

419 

221 

232 

242 

252 

263 

273 

284 

294 

3°4 

3*5 

420 

325 

335 

346 

356 

366 

377 

38v 

397 

408 

418 

421 

428 

439 

449 

459 

469 

480 

490 

500 

5" 

521 

IO 

422 

531 

542 

552 

562 

572 

583 

593 

603 

613 

624 

I 

1.0 

423 

634 

644 

655 

665 

675 

685 

696 

706 

716 

726 

2 

2.0 

424 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

3 

4 

3-0 
4.0 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

93i 

5-o 

426 

941 

95i 

961 

972 

982 

992 

*002 

*OI2 

*022 

*Q33 

7 

6.0 
7.0 

427 

63  043 

•053 

063 

073 

083 

094 

104 

114 

124 

134 

8 

8.0 

428 

144 

i55 

165 

175 

185 

195 

205 

215 

225 

236 

9 

9.0 

429 

246 

256 

266 

276 

286 

296 

306 

3J7 

327 

337 

430 

347 

357 

367 

377 

387 

397 

407 

4i7 

428 

438 

431 

448 

458 

468 

478 

488 

498 

508 

5i8 

528 

538 

432 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 

433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

434 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 

435 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 

g 

436 

949 

959 

969 

979 

988 

998 

*oo8 

*oi8 

*028 

*o38 

i 

0.9 

437 

64  048 

058 

068 

078 

088 

098 

1  08 

118 

128 

137 

2 

1.8 

438 

147 

157 

167 

177 

I87 

197 

207 

217 

227 

237 

3 

4 

2  .  7 

3-6 

439 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

I 

4-5 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

7 

5  •  4 
6.3 

441 

444 

454 

464 

473 

483 

493 

5°3 

5*3 

523 

S32 

8 

7-2 

442 

& 

552 

562 

572 

582 

59i 

601 

6n 

621 

631 

9 

8.1 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

445 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 

446 

933 

943 

953 

963 

972 

982 

992 

*002 

*OII 

*02I 

447 

65  031 

040 

050 

060 

070 

079 

089 

099 

1  08 

118 

448 

128 

137 

147 

157 

167 

176 

186 

I96 

205 

215 

449 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 

LOGARITHMS   OF  NUMBERS 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.Pts. 

450 

65  321 

33i 

341 

35° 

360 

369 

379 

389 

398 

408 

451 

418 

427 

437 

447 

456 

466 

475 

485 

495 

504 

452 

5J4 

523 

533 

543 

552 

562 

571 

581 

600 

453 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 

454 

706 

7I5 

725 

734 

744 

753 

763 

772 

782 

792 

455 

801 

8n 

820 

830 

839 

'849 

858 

868 

877 

887 

456 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 

457 

992 

*OOI 

*OII 

*020 

*O3O 

*°39 

*O49 

*o58 

*o68 

*o77 

458 

66  087 

096 

106 

"5 

124 

134 

143 

153 

162 

172 

10 

I  .  O 

459 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 

2 

2.O 

460 

276 

285 

295 

3°4 

3*4 

323 

332 

342 

351 

361 

3 

3-0 

461 

37° 

380 

389 

398 

408 

4i7 

427 

43  6 

445 

455 

4 
5 

4.0 
5-0 

462 

464 

474 

483 

492 

502 

511 

521 

53° 

539 

549 

6 

6.0 

463 

558 

567 

577 

586 

596 

605 

614 

624 

633 

642 

I 

7.0 
8.0 

464 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

9 

9.0 

465 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 

466 

-  839 

848 

857 

867 

876 

885 

894 

904 

9*3 

922 

467 

932 

94i 

95° 

960 

969 

978 

987 

997 

468 

67  025 

°34 

043 

05? 

062 

071 

080 

089 

099 

108 

469 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

47i 

302 

3" 

321 

330 

339 

348 

357 

367 

376 

385 

472 

394 

403 

422 

440 

449 

459 

468 

477 

x 

y 

o.  9 

473 

486 

495 

5°4 

5*4 

523 

532 

54i 

55° 

560 

569 

3 

1.8 

474 

578 

587 

596 

605 

614 

624 

633 

642 

651 

660 

3 

4 

2-7 

3.6 

475 

669 

679 

688 

697 

706 

7I5 

724 

733 

742 

752 

5 

4-5 

476 

761 

770 

779 

788 

797 

806 

825 

834 

843 

6 

477 

852 

861 

870 

879 

888 

897 

906 

916 

925 

934 

87 

7.2 

478 

943 

952 

961 

97° 

979 

988 

997 

*oo6 

•"015 

*024 

9 

8.1 

479 

68  034 

043 

052 

061 

070 

079 

088 

097 

106 

"5 

480 

124 

133 

142 

151 

1  60 

169 

178 

187 

196 

205 

481 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

482 

3°5 

3J4 

323 

332 

34i 

35° 

359 

368 

377 

386 

483 

395 

404 

4i3 

422 

431 

440 

449 

458 

467 

476 

484 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 

485 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 

486 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

S 

0.8 

487 
488 

753 
842 

762 
851 

771 
860 

780 
869 

878 

III 

806 
895 

815 
904 

824 

833 
922 

a 

3 

4 

1.6 
2.4 

3-2 

489 

931 

940 

949 

958 

966 

975 

984 

993 

*002 

*OII 

5 

4.0 

490 

69  020 

028 

037 

046 

°55 

064 

°73 

082 

090 

099 

o 

7 

4.8 
5  6 

491 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

8 

6.4 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

9 

7-2 

493 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 

494 

373 

381 

39° 

399 

408 

425 

434 

443 

452 

495 

461 

469 

478 

487 

496 

5°4 

513 

522 

531 

539 

496 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 

497 

636 

644 

653 

662 

671 

679 

688 

697 

7°5 

714 

498 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 

499 

8x0 

819 

827 

836 

845 

854 

8-62 

871 

880 

888 

103 


LOGARITHMS  OF   NUMBERS. 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.Ptg. 

500 

69  897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

501 

992 

OOI 

010 

018 

027 

036 

044 

°53 

062 

502 

70  070 

079 

088 

096 

105 

114 

122 

131 

140 

148 

503 

157 

165 

174 

183 

191 

200 

209 

217 

226 

234 

504 

243 

252 

260 

269 

278 

286 

295 

3°3 

312 

321 

505 

329 

338 

346 

355 

364 

372 

381 

389 

398 

406 

506 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 

507 

501 

509 

518 

526 

535 

544 

552 

569 

578 

508 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

i 

9 

o  9 

509 

672 

680 

689 

697 

706 

7U 

723 

731 

740 

749 

2 

1.8 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

3 

2.7 

i  A 

5" 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

4 
5 

3-° 
4-5 

5" 

927 

935 

944 

952 

961 

969 

978 

986 

995 

*oo3 

6 

5-4 

6m 

71  012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

I 

•  3 
7.2 

514 

096 

105 

"3 

122 

130 

139 

147 

155 

164 

172 

9 

8.1 

515 

181 

l89 

198 

2O6 

214 

223 

231 

240 

248 

257 

516 

265 

273 

282 

290 

299 

3°7 

324 

332 

341 

5*7 

349 

357 

366 

374 

383 

399 

408 

416 

425 

518 

433 

441 

45° 

458 

466 

475 

483 

492 

500 

508 

5ip 

517 

525 

533 

542 

55° 

559 

567 

575 

584 

592 

520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 

521 

684 

692 

700 

709 

717 

725 

734 

742 

75° 

759 

522 

767 

784 

792 

800 

809 

817 

825 

834 

842 

I 

o  8 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

2 

1.6 

524 

933 

941 

95° 

958 

966 

975 

983 

991 

999 

*oo8 

3 

4 

2.4 

3.  2 

525 

72  016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

5 

4.0 

526 

099 

107 

"5 

123 

132 

140 

148 

156 

165 

173 

6 

4.8 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

8 

tS 

528 

263 

272 

280 

288 

296 

3°4 

321 

329 

337 

9 

7.2 

529 

346 

354 

362 

37° 

378 

387 

395 

403 

411 

419 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 

5°9 

526 

534 

542 

55° 

558 

567 

575 

583 

532 

591 

599 

607 

616 

624 

632 

640 

648 

656 

665 

533 

673 

68  1 

689 

697 

7°5 

713 

722 

730 

738 

746 

534 

754 

763 

77° 

779 

787 

795 

803 

811 

819 

827 

535 

835 

843 

852 

860 

868 

876 

884 

892 

900 

908 

536 

916 

925 

*933 

941 

949 

*957 

965 

973 

981 

*989 

I 

7 
0.7 

537 

997 

*oo6 

*022 

*O3O 

^046 

*°54 

*062 

2 

1.4 

538 

73  °78 

086 

094 

102 

in 

119 

127 

135 

143 

151 

3 

4 

2.  I 
2.8 

539 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

5 

3-5 

540 

239 

247 

255 

263 

272 

280 

288 

296 

3°4 

312 

0 

7 

4.2 

4  9 

541 

320 

328 

336 

344 

352 

360 

368 

376 

384 

392 

8 

5-6 

542 

400 

408 

416 

424 

432 

440 

448 

456 

46^ 

472 

9 

6.3 

543 

480 

488 

496 

5°4 

512 

520 

528 

536 

544 

544 

560 

568 

576 

584 

592 

600 

608 

616 

624 

632 

545 

640 

648 

656 

664 

672 

679 

687 

695 

7°3 

711 

546 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 

547 

799 

807 

815 

823 

830 

838 

846 

854 

862 

870 

548 

878 

886 

894 

902 

910 

918 

926 

933 

94i 

949 

549 

957 

965 

973 

981 

989 

997 

*oi3 

*020 

*028 

74 

104 


LOGARITHMS   OF  NUMBERS. 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  PtB. 

550 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

551 

US 

123 

I31 

139 

147 

155 

162 

170 

178 

186 

552 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 

553 

273 

280 

288 

296 

364 

312 

320 

327 

335 

343 

554 

3Si 

359 

367 

374 

382, 

<39° 

398 

406 

414 

421 

555 

429 

437 

445 

453 

461 

468 

476 

484 

492 

500 

556 

507 

5i5 

523 

531 

539 

547 

554 

562 

57° 

578 

557 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 

558 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 

559 

741 

749 

757 

>64 

772 

780 

788 

796 

803 

811 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

56i 

896 

904 

912 

920 

927 

935 

943 

95° 

958 

966 

562 

974 

981 

989 

997 

*ooS 

*OI2 

*020 

*028 

*035 

*043 

563 

75  051 

059 

066 

074 

082 

089 

097 

i°5 

H3 

120 

I  o  8 

564 

128 

136 

143 

151 

159 

1  66 

174 

182 

189 

197 

2   1.6 

565 

205 

213 

220 

228 

236 

243 

251 

259 

266 

274 

3  2-4 
4  3-2 

566 

282 

289 

297 

3°5 

312 

320 

328 

335 

343 

351 

§4-° 

567 

358 

366 

374 

381 

389 

397 

404 

412 

420 

427 

4-8 

IT    r  f\ 

568 

435 

442 

45° 

458 

465 

473 

481 

488 

496 

5°4 

I  1:1 

569 

5" 

5i9 

526 

534 

542 

549 

557 

565 

572 

580 

9  7-2 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 

57i 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 

572 

740 

747 

755 

762 

77° 

778 

785 

793 

800 

808 

573 

8i5 

823 

831 

838 

846 

853 

861 

868 

876 

884 

574 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 

575 

967 

974 

982 

989 

997 

*oo5 

*OI2 

*020 

*027 

*Q35 

576 

76  042 

050 

°57 

065 

072 

080 

087 

°95 

103 

no 

577 

118 

125 

133 

140 

148 

155 

163 

170 

178 

185 

578 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 

579 

268 

275 

283 

290 

298 

3°5 

3J3 

320 

328 

335 

580 

343 

35° 

358 

365 

373 

380 

388 

395 

403 

410 

58i 

418 

425 

433 

440 

448 

455 

462 

47° 

477 

485 

7 

i  0.7 

582 

492 

500 

5°7 

515 

522 

53° 

537 

545 

552 

559 

2   1.4 

583 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

3  2.1 
4  2.8 

584 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

5  3-5 

585 

716 

723 

73° 

738 

745 

753 

760 

768 

775 

782 

6  4.2 

586 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

7  4-9 
8  5.6 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923 

93° 

9  6.3 

588 

938 

945 

953 

960 

967 

975 

982 

989 

997 

*oo4 

589 

77  012 

019 

026 

034 

041 

048 

056 

063 

070 

078 

590 

085 

°93 

IOO 

107 

n5 

122 

129 

137 

144 

»*$i 

59i 

i59 

1  66 

173 

181 

188 

195 

203 

210 

217 

225 

592 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 

593 

305 

3i3 

320 

327 

335 

342 

349 

357 

364 

37i 

594 

379 

386 

393 

401 

408 

415 

422 

43° 

437 

444 

595 

452 

459 

466 

474 

481 

488 

495 

5°3 

5io 

517 

596 

525 

S32 

539 

546 

554 

56l 

568 

576 

583 

59° 

597 

597 

605 

612 

619 

627 

634 

641 

648 

656 

663 

598 

.  670 

677 

685 

692 

699 

706 

714 

721 

728 

735 

599 

743 

75o 

757 

764 

772 

779 

786 

793 

801 

808 

105 


LOGARITHMS   OF   NUMBERS. 


No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

600 

77  8i5 

822 

830 

837 

844 

851 

859 

866 

873 

880 

601 

887 

895 

902 

909 

916 

924 

93i 

93s 

945 

952 

602 

960 

967 

974 

981 

988 

996 

*oo3 

*OIO 

*oi7 

*025 

603 

78  032 

°39 

046 

°53 

061 

068 

075 

082 

089 

097 

604 

104 

in 

118 

125 

132 

140 

147 

154 

161 

1  68 

605 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 

606 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 

607 

3i9 

326 

333 

340 

347 

355 

362 

369 

376 

383 

608 

39° 

398 

405 

412 

419 

426 

433 

440 

447 

455 

I 

8 

0.8 

609 

462 

469 

476 

483 

490 

497 

5°4 

512 

5*9 

526 

2 

1.6 

610 

533 

540 

547 

554 

56i 

569 

576 

583 

59° 

597 

3 

2.4 

611 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

4 
5 

3-2 
4.0 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

6 

4-8 

613 

746 

753 

760 

767 

774 

78i 

789 

796 

803 

8!0 

I 

I'" 

614 

817 

824 

831 

838 

845 

852 

859 

866 

873 

880 

9 

7.2 

615 

888 

895 

902 

909 

916 

923 

93° 

937 

944 

951 

616 

958 

965 

972 

979 

986 

993 

*000 

*oo7 

*oi4 

*02I 

617 

79  029 

036 

043 

050 

057 

064 

071 

078 

085 

092 

618 

099 

1  06 

"3 

120 

127 

i34 

141 

148 

155 

162 

619 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 

620 

239 

246 

253 

260 

267 

274 

28l 

288 

295 

302 

621 

3°9 

316 

323 

33° 

337 

344 

351 

358 

365 

372 

622 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 

I 

/ 
o.  7 

623 

449 

456 

463 

47° 

477 

484 

491 

498 

5°5 

511 

2 

i-4 

524 

5i8 

525 

S32 

539 

546 

553 

560 

567 

574 

581 

3 

4 

2.  I 
2.8 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

5 

3-5 

626 

657 

664 

671 

678 

685 

692 

699 

706 

7J3 

720 

6 

4-2 

627 

727 

734 

74i 

748 

754 

761 

768 

775 

782 

789 

I 

4.  9 
5-6 

628 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

9 

6-3 

629 

865 

872 

879 

886 

893 

900 

906 

9i3 

920 

927 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 

631 

80  003 

010 

017 

024 

030 

°37 

044 

Q51 

058 

065 

632 

072 

079 

085 

092 

099 

1  06 

113 

120 

127 

i34 

633 

140 

147 

154 

161 

1  68 

i75 

182 

1  88 

i95 

202 

634 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 

635 

277 

284 

291 

298 

305 

312 

3i8 

325 

332 

339 

6 

636 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 

I 

0.6 

637 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 

2 

1.2 

T   Q 

638 

482 

489 

496 

502 

5°9 

5i6 

523 

53° 

536 

543 

3 

4 

I  .  O 

2-4 

639 

55° 

557 

564 

57° 

577 

584 

59i 

598 

604 

611 

3-0 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

7 

3-6 

4  ^ 

641 

686 

693 

699 

706 

7J3 

720 

726 

733 

740 

747 

8 

4.8 

642 

754 

760 

767 

774 

78! 

787 

794 

801 

808 

814 

9 

5-4 

643 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

644 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

645 

956 

963 

969 

976 

983 

990 

996 

*oo3 

*OIO 

*oi7 

646 

81  023 

030 

°37 

043 

050 

057 

064 

070 

077 

084 

647 

090 

097 

104 

in 

117 

124 

131 

137 

144 

151 

648 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 

649 

224 

231 

238 

245 

25r 

258 

265 

271 

278 

28S 

106 


LOGARITHMS   OF   NUMBERS. 


No. 

o 

i 

"*) 

3 

4 

3 

6 

7 

8 

9 

Pp.  Pts. 

650 

81  291 

298 

305 

3" 

3,8 

325 

33i 

338 

345 

35i 

651 

353 

365 

37i 

378 

385 

3Qi 

398 

405 

411 

418 

652 

425 

43  1 

438 

445 

45i 

458 

465 

471 

478 

485 

653 

491 

498 

5°5 

511 

5i« 

525 

53i 

538 

544 

55i 

654 

553 

564 

57i 

578 

584 

•591 

598 

604 

611 

617 

655 

624 

631 

637 

^644 

651 

657 

664 

671 

677 

684 

656 

690 

697 

704 

710 

717 

723 

73° 

737 

743 

75° 

657 

757 

763 

77° 

776 

783 

790 

796 

803 

809 

816 

658 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

659 

889 

895 

902 

9-08 

9J5 

921 

928 

935 

94i 

948 

660 

954 

961 

968 

974 

981 

987 

994 

*ooo 

*oo7 

*oi4 

661 

82  020 

027 

°33 

040 

046 

°53 

060 

066 

°73 

079 

662 

086 

092 

099 

i°5 

112 

119 

125 

132 

138 

145 

663 

I51 

158 

164 

171 

I78 

184 

191 

197 

204 

2IO 

7 
I  o.  7 

664 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

2  1.4 

665 

282 

289 

295 

302 

308 

3i5 

321 

328 

334 

341 

3  2.1 

4  2.8 

666 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

5  3-5 

667 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

o  4.2 

668 

478 

484 

491 

497 

5°4 

510 

5i7 

523 

53° 

536 

7  4-9 
8  5.6 

669 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

9  6-3 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

671 

672 

679 

685 

692 

698 

7°5 

711 

718 

724 

73° 

672 

737 

743 

75° 

756 

763 

769 

776 

782 

789 

795 

673 

802 

808 

814 

821 

827 

834 

840 

847 

853 

860 

674 

866 

872 

879 

885 

892 

898 

9°5 

911 

918 

924 

675 

93° 

937 

943 

95° 

956 

963 

969 

975 

982 

988 

676 

995 

*OOI 

*oo8 

*oi4 

*020 

*027 

*033 

*040 

*046 

*052 

677 

83  °59 

065 

072 

078 

085 

091 

097 

104 

no 

117 

678 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 

679 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 

680 

25i 

257 

264 

270 

276 

283 

289 

296 

302 

308 

681 

3*5 

321 

327 

334 

340 

347 

353 

359 

366 

372 

I  o.  6 

682 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

2   1.2 

683 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

3  1-8 
4  2-4 

684 

506 

512 

5i8 

525 

531 

537 

544 

55° 

556 

563 

5  3-o 

685 

569 

575 

582 

588 

594 

601 

607 

613 

620 

626 

6  3.6 

68*6 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

7   4-2 

8  4-8 

687 

696 

702 

708 

7i5 

721 

727 

734 

740 

746 

753 

9  5-4 

688 

759 

765 

771 

778 

784 

790 

797 

803 

809 

816 

689 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

691 

948 

954 

960 

967 

973 

979 

985 

992 

998 

*oo4 

692 

84  on 

017 

023 

029 

036 

042 

048 

055 

061 

067 

- 

693 

°73 

080 

086 

092 

098 

i°5 

in 

117 

123 

130 

694 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 

695 

198 

205 

211 

217 

223 

230 

236 

•  242 

248 

255 

696 

261 

267 

273 

280 

286 

292 

298 

305 

3" 

317 

697 

323 

33° 

336 

342 

348 

354 

361 

367 

373 

379 

698 

386 

392 

398 

404 

410 

4i7 

423 

429 

435 

442 

699 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 

107 


LOGARITHMS  OF  NUMBERS. 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

70O 

84  510 

5i6 

522 

528 

535 

54i 

547 

553 

559 

566 

701 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 

702 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 

703 

696 

702 

708 

714 

720 

726 

733 

739 

745 

75i 

704 

757 

763 

77° 

776 

782 

788 

794 

800 

807 

813 

70S 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 

706 

880 

887 

893 

899 

9°5 

911 

917 

924 

93° 

936 

707 

942 

948 

954 

960 

967 

973 

979 

985 

991 

997 

708 

85  003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

I 

7 
o.  7 

709 

065 

071 

077 

083 

089 

°95 

101 

107 

114 

120 

2 

1.4 

710 

126 

132 

138 

144 

i5° 

156 

163 

169 

175 

181 

3 

2.1 

rt   Q 

711 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

4 
5 

2  .  o 

3-5 

712 

248 

254 

260 

266 

272 

278 

285 

291 

297 

3°3 

6 

4.2 

713 

3°9 

315 

321 

327 

333 

339 

345 

352 

358 

364 

I 

4-9 
5.6 

714 

37o 

376 

382 

388 

394 

400 

406 

412 

418 

425 

9 

6.3 

715 

43i 

437 

443 

449 

455 

461 

467 

473 

479 

485 

716 

49  1 

497 

503 

5°9 

5i6 

522 

528 

534 

540 

546 

717 

552 

558 

564 

57° 

576 

582 

588 

594 

600 

606 

718 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 

719 

673 

679 

685 

691 

697 

7°3 

709 

7i5 

721 

727 

720 

733 

739 

745 

75i 

757 

763 

769 

775 

781 

788 

721 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 

£. 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

J 

u 

0.6 

723 

914 

920 

926 

932 

938 

944 

95° 

956 

962 

968 

2 

1.2 

724 

974 

980 

986 

992 

998 

*oo4 

*OIO 

*oi6 

*022 

*028 

3 

4" 

1.8 

2  ,  A 

725 

86  034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

1 

5 

3.0 

726 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

6 

3-6 

727 

153 

*59 

165 

171 

177 

183 

189 

195 

201 

207 

I 

y 

728 

213 

219 

225 

23I 

237 

243 

249 

255 

26l 

267 

9 

5-4 

729 

273 

279 

285 

29I 

29.7 

3°3 

308 

3i4 

320 

326 

730 

S32 

338 

344 

35° 

356 

362 

368 

374 

380 

386 

731 

392 

398 

404 

410 

4i5 

421 

427 

433 

439 

445 

732 

45i 

457 

463 

469 

475 

481 

487 

493 

499 

5°4 

733 

5i° 

5i6 

522 

528 

534 

540 

546 

552 

558 

564 

734 

57° 

576 

58i 

587 

593 

599 

605 

611 

617 

623 

735 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 

736 

688 

694 

700 

7°5 

711 

717 

723 

729 

735 

74i 

I 

5 

0.5 

737 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

2 

I.O 

738 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

3 

4 

i-5 

2.0 

739 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

2-5 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

»7 

3-0 

3r 

74i 

982 

988 

994 

999 

*oo5 

*OII 

*oi7 

*023 

*029 

*°35 

8 

•  J 

4.0 

742 

87  040 

046 

052 

058 

064 

070 

°75 

081 

087 

°93 

9 

4-5 

743 

099 

I05 

in 

116 

122 

128 

134 

140 

146 

151 

744 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

745 

216 

221 

227 

233 

239 

245 

25i 

256 

262 

268 

746 

274 

280 

286 

291 

297 

3°3 

3°9 

3*5 

320 

326 

747 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 

748 

39° 

396 

402 

408 

4i3 

419 

425 

43i 

437 

442 

749 

448 

454 

460 

466 

47i 

477 

483 

489 

495 

500 

108 


LOGARITHMS  OF  NUMBERS. 


No. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.Pt8. 

750 

87  506 

512 

518 

523 

529 

535 

54i 

547 

SS2 

558 

751 

564 

57° 

576 

58i 

587 

593 

599 

604 

610 

616 

752 

622 

628 

633 

639 

645 

651 

656 

662 

668 

674 

753 

679 

685 

691 

697 

te 

708 

714 

720 

726 

73i 

754 

737 

743 

749 

754 

76*. 

766 

772 

777 

783 

789 

755 

795 

800 

806* 

812 

818 

823 

829 

835 

841 

846 

756 

852 

858 

864 

869 

875 

88  1 

887 

892 

898 

904 

757 

910 

915 

921 

927 

933 

938 

944 

95° 

955 

961 

758 

967 

973 

978 

984 

990 

996 

*OOI 

*oo7 

*oi3 

*oi8 

759 

88  024 

030 

036 

041 

047 

053 

058 

064 

070 

076 

760 

081 

087 

°93 

098 

104 

no 

116 

121 

127 

133 

761 

138 

144 

15° 

156 

161 

167 

173 

I78 

184 

190 

762 

*95 

2OI 

207 

213 

218 

224 

230 

235 

241 

247 

763 

252 

258 

264 

270 

275 

281 

287 

292 

298 

3°4 

6 

I  o.  6 

764 

3°9 

315 

321 

326 

332 

338 

343 

349 

355 

360 

2   1.2 

765 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

3  1.8 
4  2.4 

766 

423 

429 

434 

440 

446 

45i 

457 

463 

468 

474 

5  3-o 

767 

480 

485 

491 

497 

502 

508 

5*3 

5*9 

525 

53° 

6  3.6 

768 

536 

542 

547 

553 

559 

564 

57° 

576 

58i 

587 

7  4-2 
8  4.8 

769 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 

9  5-4 

770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

771 

705 

711 

717 

722 

728 

734 

739 

745 

75° 

756 

772 

762 

767 

773 

779 

784 

79° 

795 

801 

807 

812 

773 

8x8 

824 

829 

835 

840 

846 

852 

857 

863 

868 

774 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 

775 

93° 

936 

941 

947 

953 

958 

964 

969 

975 

981 

776 

986 

992 

997 

*oo3 

*oo9 

*oi4 

*020 

*025 

*o3i 

*037 

777 

89  042 

048 

°53 

°59 

064 

070 

076 

081 

087 

092 

778 

098 

104 

109 

"5 

I2O 

126 

I3I 

137 

143 

148 

779 

154 

*59 

165 

170 

I76 

182 

187 

193 

198 

204 

780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

781 

265 

271 

276 

282 

287 

293 

298 

3°4 

310 

3i5 

5 

I  0.5 

782 

321 

326 

332 

337 

343 

348 

354 

360 

365 

37i 

2   1.0 

783 

376 

382 

387 

393 

398 

404 

409 

4i5 

421 

426 

3  i.S 

4  2.0 

784 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

5  2.5 

785 

487 

492 

498 

5°4 

5°9 

5i5 

520 

526 

53i 

537 

6  3.0 

786 

542 

548 

553 

559 

564 

57° 

575 

58i 

586 

592 

7  3-5 
8  4.0 

787 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

9  4-5 

788 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

789 

708 

7i3 

719 

724 

73° 

735 

741 

746 

752 

757 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

791 

818 

823 

829 

834 

840 

845 

851 

856 

862 

867 

792 

873 

878 

883 

889 

894 

900 

9°5 

911 

916 

922 

793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 

794 

982 

988 

993 

998 

*oo4 

*009 

*oi5 

*O2O 

*026 

*°3! 

795 

90  037 

042 

048 

053 

059 

064 

069 

075 

080 

086 

796 

091 

097 

102 

108 

H3 

119 

124 

129 

135 

140 

797 

146 

151 

157 

162 

168 

i73 

179 

184 

189 

195 

798 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 

799 

255 

260 

266 

271 

276 

282 

287 

293 

298 

3°4 

109 


LOGARITHMS  OF  NUMBERS. 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.Pts. 

800 

90  309 

3i4 

320 

325 

33i 

336 

342 

347 

352 

358 

801 

363 

369 

374 

380 

385 

39° 

396 

401 

407 

412 

802 

417 

423 

428 

434 

439 

445 

45° 

455 

461 

466 

803 

472 

477 

482 

488 

493 

499 

5°4 

5°9 

5J5 

520 

804 

526 

53i 

536 

542 

547 

553 

558 

563 

569 

574 

805 

580 

585 

59° 

596 

601 

607 

612 

617 

623 

628 

806 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 

807 

687 

693 

698 

7°3 

709 

714 

720 

725 

73° 

736 

808 

741 

747 

752 

757 

763 

768 

773 

779 

784 

789 

809 

795 

800 

806 

8n 

816 

822 

827 

832 

838 

843 

810 

849 

854 

%9 

865 

870 

875 

881 

886 

891 

897 

811 

902 

907 

9i3 

918 

924 

929 

934 

940 

945 

95° 

812 

956 

961 

966 

972 

977 

982 

988 

993 

998 

*oo4 

813 

91  009 

014 

020 

025 

030 

036 

041 

046 

052 

°57 

I  o.  6 

814 

062 

068 

°73 

078 

084 

089 

094 

IOO 

i°5 

no 

2   1.2 

8i5 

116 

121 

126 

132 

i37 

142 

148 

153 

158 

164 

3  1.8 
4  2.4 

816 

169 

'74 

180 

185 

190 

196 

201 

206 

212 

217 

5  3-0 

817 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270 

6  3-6 

818 

275 

28l 

286 

291 

297 

302 

3°7 

312 

3^ 

323 

*7  4-2 
8  4.8 

819 

328 

334 

339 

344 

35° 

355 

360 

365 

371 

376 

9  5-4 

820 

38l 

387 

392 

397 

403 

408 

4i3 

418 

424 

429 

821 

434 

440 

445 

45° 

455 

461 

466 

471 

477 

482 

822 

487 

492 

498 

5°3 

508 

5*4 

5*9 

524 

529 

535 

823 

540 

545 

55i 

556 

56i 

566 

572 

577 

582 

587 

824 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 

825 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 

826 

698 

7°3 

709 

714 

719 

724 

73° 

735 

740 

745 

827 

75i 

756 

761 

766 

772 

777 

782 

787 

793 

798 

828 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 

829 

855 

861 

866 

871 

876 

882 

887 

892 

897 

903 

830 

908 

9*3 

918 

924 

929 

934 

939 

944 

95° 

955 

831 

960 

965 

971 

976 

981 

986 

991 

997 

*002 

*oo7 

5 

In  c 

832 

92  012 

018 

023 

028 

033 

038 

044 

049 

054 

°59 

°*  5 
2   1.0 

833 

065 

070 

075 

080 

085 

091 

096 

101 

106 

in 

3  1-5 

A   2  .0 

834 

117 

122 

127 

132 

137 

143 

148 

J53 

158 

163 

5  2\s 

835 

I69 

174 

179 

184 

189 

195 

200 

205 

210 

2I5 

o  3.0 

836 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

7  3-5 
8  4.0 

837 

273 

278 

283 

288 

293 

298 

3°4 

3°9 

3*4 

3J9 

9  4-5 

838 

324 

33° 

335 

340 

345 

350 

355 

361 

366 

37i 

839 

376 

38l 

387 

392 

397 

402 

407 

412 

418 

423 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

841 

480 

485 

49° 

495 

500 

5°5 

511 

5i6 

52i 

526 

842 

531 

536 

542 

547 

552 

557 

562 

567 

572 

578 

843 

583 

588 

593 

598 

603 

609 

614 

619 

624 

629 

844 

634 

639 

645 

650 

655 

660 

665 

670 

675 

68  1 

845 

686 

691 

696 

701 

706 

711 

716 

722 

727 

732 

846 

737 

742 

747 

752 

758 

763 

768 

773 

778 

783 

847 

788 

793 

799 

804 

809 

814 

819 

824 

829 

834 

848 

840 

845 

850 

855 

860 

865 

870 

875 

881 

886 

849 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 

no 


LOGARITHMS   OF   NUMBERS. 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp 

.Pts. 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

851 

993 

998 

*003 

*oo8 

*OI3 

*oi8 

*024 

*029 

*°34 

*Q39 

852 

93  °44 

049 

054 

°59 

064 

069 

°75 

080 

085 

090 

853 

°95 

100 

i°5 

no 

1  151' 

120 

125 

131 

136 

141 

854 

146 

I5i 

156 

161 

1  66 

I7I 

176 

181 

186 

192 

855 

197 

202 

207 

ai2 

217 

222 

227 

232 

237 

242 

856 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 

857 

298 

3°3 

308 

313 

3i8 

323 

328 

334 

339 

344 

858 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

I 

6 

0.6 

859 

399 

404 

409 

414 

420 

425 

43° 

435 

440 

445 

2 

1.2 

860 

45° 

455 

460 

465 

47° 

475 

480 

485 

49° 

495 

3 

1.8 

86  1 

500 

5°5 

5i° 

5J5 

520 

526 

53i 

536 

54i 

546 

4 

5 

2.4 

3  -° 

862 

551 

556 

56i 

566 

57i 

576 

58i 

586 

591 

596 

6 

3-6 

863 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

7 
g 

4.2 

A  8 

864 

651 

.656 

661 

666 

671 

676 

682 

687 

692 

697 

9 

4  •  ° 
5-4 

865 

702 

707 

712 

717 

722 

727 

732 

737 

742 

747 

866 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 

867 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 

868 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 

869 

902 

907 

912 

917 

922 

927 

932 

937 

942 

947 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

871 

94  002 

007 

012 

017 

022 

027 

032 

037 

042 

047 

872 

052 

°57 

062 

067 

072 

077 

082 

086 

091 

096 

j 

5 

QB  5 

873 

101 

1  06 

III 

116 

121 

126 

131 

136 

141 

146 

2 

I.O 

874 

151 

156 

161 

1  66 

171 

176 

181 

186 

191 

196 

3 

i.S 

2  .  O 

875 

2OI 

206 

211 

216 

221 

226 

231 

236 

240 

245 

5 

2.5 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

6 

3-0 

877 

300 

3°5 

310 

3i5 

320 

325 

33° 

335 

340 

345 

I 

3-5 
4.0 

878 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

9 

4-5 

879 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

881 

498 

5°3 

5°7 

512 

517 

522 

527 

532 

537 

542 

882 

547 

552 

557 

562 

567 

57i 

576 

58i 

586 

59i 

883 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 

884 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 

885 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 

886 

743 

748 

753 

758 

763 

768 

^773 

778 

783 

787 

X 

4 

0.4 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

a 

0.8 

888 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

3 

4 

.2 

.6 

889 

890 

895 

900 

9°5 

910 

9i5 

919 

924 

929 

934 

5 

.0 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

o 

w 

.4 
.8 

891 

988 

993 

998 

*002 

*oo7 

*OI2 

*oi7 

*022 

*027 

*032 

8 

3-2 

892 

95  °36 

041 

046 

051 

056 

061 

066 

071 

075 

080 

9 

3-6 

893 

085 

090 

'  °95 

100 

105 

109 

114 

119 

124 

129 

894 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

895 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 

896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

897 

279 

284 

289 

294 

299 

3°3 

308 

3i3 

318 

323 

898 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 

899 

376 

381 

386 

39° 

395 

400 

405 

410 

415 

419 

III 


LOGARITHMS  OF   NUMBERS. 


No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

QOO 

95  424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

QOI 

472 

477 

482 

487 

492 

497 

5°i 

506 

5n 

5i6 

902 

52i 

525 

53° 

535 

540 

545 

55° 

554 

559 

564 

903 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 

904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

90S 

665 

670 

674 

679 

684 

689 

694 

698 

7°3 

708 

906 

713 

718 

722 

727 

732 

737 

742 

746 

75i 

756 

907 

761 

766 

77° 

775 

780 

785 

789 

794 

799 

804 

908 

809 

8i3 

818 

823 

828 

832 

837 

842 

847 

852 

909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

9IO 

904 

909 

914 

918 

923 

928 

933 

938 

942- 

947 

911 

9S2 

957 

961 

966 

971 

976 

980 

985 

990 

995 

912 

999 

*oo4 

*oo9 

*oi4 

*oi9 

*O23 

*028 

*<>33 

*038 

*042 

913 

96  047 

052 

°57 

061 

066 

071 

076 

080 

085 

090 

i    0.5 

914 

°95 

099 

104 

109 

114 

118 

123 

128 

133 

137 

2      1.0 

915 

142 

147 

J52 

156 

161 

1  66 

171 

175 

180 

185 

3     i.S 

4      2.0 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

'232 

5     2.5 

917 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

o    3.0 

•7       t    r 

9l8 

284 

289 

294 

298 

3°3 

308 

3i3 

3i7 

322 

327 

7     3-5 
o     4.0 

919 

S32 

336 

34i 

346 

35° 

355 

360 

365 

369 

374 

9     4-5 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

921 

426" 

43  1 

435 

440 

445 

45° 

454 

459 

464 

468 

922 

473 

478 

483 

487 

492 

497 

5oi 

506 

5n 

5i5 

923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

924 

567 

572 

577 

58i 

586 

59i 

595 

600 

605 

609 

925 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 

920 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

927 

708 

713 

717 

722 

727 

73i 

736 

74i 

745 

75° 

928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

931 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

4 

104 

932 

942 

946 

95i 

956 

960 

965 

97° 

974 

979 

984 

208 

933 

988 

993 

997 

*002 

*oc>7 

*OII 

*oi6 

*O2I 

*025 

*030 

3     i   2 
416 

934 

97  035 

°39 

044 

049 

°53 

058 

063 

067 

072 

077 

520 

935 

081 

086 

090 

095 

IOO 

104 

109 

114 

118 

123 

624 

<7        0    X 

936 

128 

132 

137 

142 

146 

151 

155 

1  60 

165 

169 

725 

8     3  2 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

9     3-6 

938 

32O 

225 

230 

234 

239 

243 

248 

253 

257 

262 

939 

267 

271 

276 

280 

285 

290 

294 

299 

3°4 

308 

940 

3*3 

3i7 

322 

327 

331 

336 

340 

345 

35° 

354 

941 

359 

364 

368 

373 

377 

382 

387 

39i 

396 

400 

942 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

943 

45i 

45  6 

460 

465 

47° 

474 

479 

483 

488 

493 

944 

497 

502 

506 

5n 

5i6 

520 

525 

529 

534 

539 

945 

543 

548 

SS2 

557 

562 

566 

57i 

575 

580 

585 

946 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

947 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 

948 

681 

685 

690 

695 

699 

704 

708 

7i3 

717 

722 

949 

727 

73i 

736 

740 

745 

749 

754 

759 

763 

768 

1X2 


LOGARITHMS  OF  NUMBERS. 


Ko. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Dp.PtB. 

95<> 

97  772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

951 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 

952 

864 

868 

873 

877 

882 

886 

891 

896 

900 

9°5 

953 

909 

914 

918 

923 

928 

932 

937 

94i 

946 

95o 

954 

955 

959 

964 

968 

9/3 

978 

982 

987 

991 

996 

955 

98  ooo 

005 

009 

014 

019 

..023 

028 

032 

°37 

041 

956 

046 

050 

°55 

*°59 

064 

068 

°73 

078 

082 

087 

957 

091 

096 

100 

i°5 

109 

114 

118 

123 

127 

132 

958 

137 

14*1 

146 

15° 

155 

159 

164 

168 

173 

177 

959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

961 

272 

277 

281 

286 

290 

295 

299 

304 

308 

3i3 

962 

318 

322 

327 

33i 

336 

340 

345 

349 

354 

358 

963 

363 

367 

372 

376 

381 

385 

39° 

394 

399 

403 

5 

i  0.5 

964 

408 

412 

417 

421 

426 

43° 

435 

439 

444 

448 

2   1.0 

965 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

3  i-5 

4  2.0 

966 

498 

502 

5°7 

5" 

5i6 

520 

525 

529 

534 

538 

5  2-5 

967 

543 

547 

552 

556 

56i 

565 

57° 

574 

579 

583 

o  3.0 

968 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

7  3-5 
8  4-0 

969 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 

9  4-5 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

971 

722 

726 

73i 

735 

740 

744 

749 

753 

758 

762 

972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

973 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

975 

900 

9°5 

909 

914 

918 

923 

927 

932 

936 

941 

976 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

977 

989 

994 

998 

*oo3 

*oo7 

*OI2 

*oi6 

*02I 

*025 

*O29 

978 

99  °34 

038 

043 

047 

052 

056 

061 

065 

069 

074 

979 

078 

083 

087 

092 

096 

IOO 

i°5 

109 

114 

118 

980 

123 

127 

I3I 

136 

140 

145 

149 

154 

158 

162 

981 

167 

171 

176 

180 

185 

189 

193 

198 

2O2 

207 

4 

I  0.4 

982 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

2   0.8 

983 

255 

260 

264 

269 

273 

277 

282 

286 

29I 

295 

3  1.2 

4  1.6 

984 

300 

3°4 

308 

313 

3i7 

322 

326 

33° 

335 

339 

5  2.0 

985 

344 

348 

352 

357 

361 

366 

37° 

374 

379 

383 

6  2.4 
7  o  a 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

/ 
8  3-2 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

47i 

9  3-6 

988 

476 

480 

484 

489 

493 

498 

502 

506 

5ii 

5i5 

989 

520 

524 

528 

533 

537 

542 

546 

55° 

555 

559 

990 

564 

568 

572 

577 

58i 

585 

59° 

594 

599 

603 

991 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 

992 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 

993 

695 

699 

704 

708 

712 

717 

721 

726 

73° 

734 

994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

995 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 

996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

997 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 

998 

9i3 

917 

922 

926 

93° 

935 

939 

944 

948 

952 

999 

957 

961 

965 

97° 

974 

978 

983 

987 

991 

996 

APPENDIX  A 


The  following  notes  and  tables  relating  to  drill  capacities 
and  losses  due  to  valves,  elbows  and  tees  are  taken  from  the 
Ingersoll-Rand  catalog. 

DRILL  CAPACITY  TABLES 

The  following  tables  are  to  determine  the  amount  of  free 
air  required  to  operate  rock  drills  at  various  altitudes  with 
air  at  given  pressures. 

The  tables  have  been  compiled  from  a  review  of  a  wide 
experience  and  from  tests  run  on  drills  of  various  sizes.  They 
are  intended  for  fair  conditions  in  ordinary  hard  rock,  but 
owing  to  varying  conditions  it  is  impossible  to  make  any 
guarantee  without  a  full  knowledge  of  existing  conditions. 

In  soft  material  where  the  actual  time  of  drilling  is  short, 
more  drills  can  be  run  with  a  given  sized  compressor  than 
when  working  in  hard  material,  when  the  drills  would  be 
working  continuously  for  a  longer  period,  thereby  increasing 
the  chance  of  all  the  drills  operating  at  the  same  time. 

In  tunnel  work,  where  the  rock  is  hard,  it  has  been  the 
experience  that  more  rapid  progress  has  been  made  when  the 
drills  were  operated  under  a  high  air  pressure,  and  that  it 
has  been  found  profitable  to  provide  compressor  capacity  in 
excess  of  the  requirements  by  about  25  per  cent.  There  is 
also  a  distinct  advantage  in  having  a  compressor  of  large 
capacity,  in  that  it  saves  the  trouble  and  expense  of 
moving  the  compressor  as  the  work  progresses,  and  will 
not  interfere  with  the  progress  of  the  work  by  crowding  the 
tunnel. 

No  allowance  has  been  made  in  the  tables  for  loss  due  to 
leaky  pipes,  or  for  transmission  loss  due  to  friction,  but  the 
capacities  given  are  merely  the  displacement  required,  so 

114 


APPENDIX  A 


115 


that  when  selecting  a  compressor  for  the  work  required  these 
matters  must  be  taken  into  account. 

Table  I  gives  cubic  feet  of  free  air  required  to  operate  one 
drill  of  a  given  size  and  under  a  given  pressure. 

Table  II  gives  multiplication  factors  for  altitudes  and 
number  of  drills  by  which  the  air  consumption  of  one  drill 
must  be  multiplied  in  order  td  give  the  total  amount  of  air. 


TABLE  I.  — CUBIC  FEET  OF  FREE  AIR  REQUIRED  TO  RUN 
ONE  DRILL  OFVHE  SIZE  AND  AT  THE  PRESSURE 
STATED  BELOW 


1 

SIZE  AND  CYLINDER  DIAMETER  OF  DRILL 

§£ 

A35 

A32 

B 

C 

D 

D 

D 

E 

F 

F 

G 

H 

H9 

o 

2" 

21" 

21" 

2f" 

3" 

31" 

3ft" 

31" 

31" 

3f" 

4f' 

5" 

51" 

60 

50 

60 

68 

82 

90 

95 

97 

100 

108 

113 

130 

150 

164 

70 

56 

68 

77 

93 

102 

108 

110 

113 

124 

129 

147 

170 

181 

80 

63 

76 

86 

104 

114 

120 

123 

127 

131 

143 

164 

190 

207 

90 

70 

84 

95 

115 

126 

133 

136 

141 

152 

159 

182 

210 

230 

100 

77 

92 

104 

126 

138 

146 

149 

154 

166 

174 

199 

240 

252 

116 


APPENDIX  A 


DETERMINE  CAPACITY  OF  COMPRESSOR  REQUIRED  TO  OPERATE 
K  DRILLS  AT  ALTITUDES  COMPARED  WITH  SEA  LEVEL 


1? 

li 

f! 


8 


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OQOi-H 


00 
»-l»O 


t^OOOOOOQOOOOSOSOSO 


I-H          O5  OS  OO  OO 

OO  OO  OO  OO  O5 


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1 


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l 


ft 


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llillfl 

.*-i   —   hn  S  <4-i  --H   .-. 


TS.fe'O 


APPENDIX    A  117 


GLOBE  VALVES,   TEES  AND  ELBOWS 

The  reduction  of  pressure  produced  by  globe  valves  is  the 
same  as  that  caused  by  the  following  additional  lengths 
of  straight  pipe,  as  calculated  by  the  formula : 

.    .  114  X  diameter  of  pipe 

Additional  length  of  pipe  =  — ; 77— 

1  +  (36  -r-  diameter) 

Diameter  of  pipe  i    1     1J     2  2}     3  3£     4      5       6  inches 

Additional  length  (2^4       7  10     13  16     20     28     36  feet 

7     8     10  12     15  18     20     22     24  inches 

44  53     70  88  115  143  162  181  200  feet 

The  reduction  of  pressure  produced  by  elbows  and  tees  is 
equal  to  two-thirds  of  that  caused  by  globe  valves.  The 
following  are  the  additional  lengths  of  straight  pipe  to  be 
taken  into  account  for  elbows  and  tees.  For  globe  valves 
multiply  by  f . 

Diameter  of  pipe }    1  1£     2  2£  3  3J  4  56  inches 

Additional  length  J    2  3      5  7  9  11  13  19    24  feet 

7  8     10  12  15  18  20  22     24  inches 

30  35    47  59  77  96  108  120  134  feet 

These  additional  lengths  of  pipe  for  globe  valves,  elbows 
and  tees  must  be  added  in  each  case  to  the  actual  lengths 
of  straight  pipe.  Thus  a  6-inch  pipe,  500  feet  long,  with 
1  globe  valve,  2  elbows  and  3  tees,  would  be  equivalent  to 
a  straight  pipe  500  +  36  +  (2  X  24)  +  (3  X  24)  =  656  feet 
long. 


APPENDIX  B 


In  the  following  tables  are  collected  all  the  reliable  data 
that  the  author  has  been  able  to  -find  relative  to  friction  in 
air  pipes. 

In  these  tables  the  significance  of  the  symbols  is  as  fol- 
lows: 

No  =  Reference  number  of  the  experiment. 
PI  =  Absolute  pressure  at  first   station   on  the  pipe  = 

pounds  per  square  inch. 

pz  =  Absolute  pressure  at  second  station  on  the  pipe  = 
pounds  per  square  inch. 

pm  =  P1       ^2  =  mean  pressure  in  pipe  between  stations. 

/  =  PI  —  Pz  =  pressure  lost  between  stations  =  pounds 

per  square  inch. 

r  =  Mean  ratio  of  compression  between  stations. 
va  =  Cubic  feet  of  free  air  passing  per  second. 
vm  =  Cubic  feet  of  compressed  air  passing  per  second. 
s  =  Velocity  of  air  in  pipe  =  feet  per  second. 
Q  =  Weight  in  pounds  of  air  passing  per  second. 
d  =  Diameter  of  pipe  in  inches. 

I  =  Length  of  pipe  in  feet. 

c  =  Coefficient  in  formula  (20),  Art.  23,viz.,f  =  c  ~  ^- 


118 


APPENDIX   B 


119 


oo 

•  *H 


oo  co  °o  o 

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-HOC  —  0 

^H  CM   T-H   ^1 


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<M(N^-I^H^-lT-H^-((M^Hr-I^H        r-(        <Mrt 


O       OS^H 
iO      t^OO 


CO      CM  I-H 
O       CT>  CO 

locor^r^oot^oor^oot-t-    i>-     cot^ 


OOOCO'-H-^CXDOOiOtO      1C      OO 
<Mr^COOlOCOCOr-CO(M!M       O       COO 


'—  1       CM      CMO 


*—  i  *—  *  O  l~~'  '•^  O      O 


•^  CO  O  •«*  —  i 

§00  oo  co  o 
O  O  CO  CO 


<M    ^H    T-l   <M    i-H    l-H 


o  t^  -^ 

CO  IO  ^H 


SSS5SS2S 

»O  •*  CO  •*  CO  CO 


OS  CO  O5  CO  CO  "tf 
CM  CM  t^  CO  O  O 


«O  CO  CM  CO  I-H  .-H 


O 
CO 


CO  CO  CO  1^-  O  CO 
OO  CO  O  t^  CO  O 


.-H  CM  CO  •<*<  O  CO 


120 


APPENDIX   B 


O5  T-H  0 

•  <M  <r> 
•^  co  oo 


Q> 


APPENDIX  B 


121 


CO  OS  CO 

II  II  II 


QOOOQOO5O5O5O5O5O5O500 
O5O>O^O5CT>O5O5OiO5O5O5 


IOO5'—  IO5OC^ 
«OCOC<I^l'^l'-H 


T—  (C<ICO'^t|iOCOt^-OOO5O'—  i 


rt<  CO  50  ^H  (M  lO 
»O  <M  t^  OC  "*  »O 
CO  t^.  t^  <£>  t^  t^ 

O  O  O  O  O  O 


<N  00  CO  00  t-  O 

i— I  CO  OO  O$  ^H  CO 

I-H  O  O  i-i  O  C5 


CO  T—  I  C<l  ^>  O  C<l 


CO  CO  CO  CO  CO 


O5   00   00    r-H    OS 

Tt<  O  CO  C^  t^ 


00  1-1  «O  IO  O  •* 


CO  OS  CO  <M  CO 

0<NSSS 


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«O  CO  »O  lO  •* 


-—  i  <rq  co  •*  10  «o 


APPENDIX  C 


During  1910  and  1911,  an  extensive  series  of  experiments 
were  made  at  Missouri  School  of  Mines  to  determine  the 
laws  of  friction  of  air  in  pipes  under  three  inches  in  diam- 
eter; the  chief  object  being  to  determine  the  coefficient 

(See  Art.  23.) 

The  general  scheme  is  illustrated  in  Fig.  15,  in  which  the 
parts  are  lettered  as  follows: 


"c"  : 


I  v<? 
in  the  formula  /  =  c  -rb  —  • 


(d) 


(f) 


Fig.  15.     Diagram  Illustrating  Assembled  Apparatus. 

o,  is  the  compressed-air  supply  pipe. 

6,  a  receiver  of  about  25  cubic  feet  capacity. 

c,  a  thermometer  set  in  receiver. 

d  and  d,  points  of  attachment  of  differential  gauge. 

/  and  /,  lengths  of  straight  pipe  going  to  and  from  the 
group  of  fittings. 

e,  the  pressure  gauge. 

g,  the  group  of  fittings  —  varied  in  different  experiments. 

ht  the  throttle  valve  to  control  pressure. 

/,  the  orifice  drum  for  measuring  air,  with  the  attach- 
ments as  in  Fig.  7. 

122 


APPENDIX  C  123 

On  each  set  of  fittings  there  were  made  ten  or  twelve  runs 
with  varying  pressures  and  quantities  of  air  in  order  to  show 

Va2 

the  relation  of  /  to  —  over  as  wide  a  field  as  possible. 

The  data  of  each  run  ,was  worked  up  and  recorded  in 
tabular  form.  Three  of  the^e  tables,  relating  to  1-inch  pipe 
and  fittings,  are  shown  herewith  as  example.  It  should 
be  recorded  that  in  the  series  of  runs  and  checks  some 
puzzling  inconsistencies  developed,  but  not  more  notice- 
able than  appears  in  the  data  from  European  experiments 
on  larger  pipe.  (See  Appendix  B.) 

In  these  tables  the  symbols  are  as  follows: 

z  =  Head,  in  inches  of  mercury,  in  differential  gauge. 

/  =  Lost  pressure  in  pounds  per  square  inch. 
pz  =  Gauge  pressure  at  entrance  to  pipe. 
rm  =  Mean  ratio  of  compression  in  pipe. 

i  =  Water  head,  in  inches,  in  U  tube  on  orifice  drum. 
Tc  =  Temperature  (centigrade)  in  drum. 
d0  =  Diameter,  in  inches,  of  orifice  in  drum. 
va  =  Volume  of  free  air  passing  (cubic  feet  per  second). 

S  =  Velocity  of  compressed  air  in  pipe  (feet  per  second) . 
/'  =  Value  of  /  when  corrected  for  temperature. 


124 


APPENDIX   C 


co 


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S  I 


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'r1     02 
Q     t«c 


r-i  CO  "tf  O  i-t  <N  T-  O  i-  O  O  T-  O  O  O  O  O  i- 


CsflOCOl^-'<*(rfCiCOCs;I 


CO^tl 


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CO-^T-iOiCO(N^tOCOOiCi 
i—  t  !>•  i—  iCOOiOCOCXD'—  iCOCD 

'—  i<Ni-HO<MOO'-HOOi—  lOO 


APPENDIX  C 


125 


M 

i  H 
« 1 

B  g  ^ 

M    >4    *d 


SJ  5*vl 

w  ^    -E 

hH  f-rij          ^* 

g  O    '5 

P  05 

III 

s  a  £ 


w 


w 


g  s 


H 


Cs}COt^C<JO-<tlOi—  i  CO  O  r-  i  C<1  O  O  O<  O  O  I-H 


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to  O  CO  <N  "3  i-H  <N  CO  T-H  (M  <N  1-1  I-H  <N        .-H 


t^.  I-H  00  CO  OS  OO  I~H  iO  l**»  ^O  '^ 
Oi  Ol  I~H  CO  t^*  *•"!  t^*  t^*  OO  t^*  CO 


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i—  i  <N  CO  Tj<  iO  CD  t>-  00  O5  O  I-H  (M  CO  Tt<  ^O  CO  1>  00 


126 


APPENDIX  C 


OO-LOio-HCOOO-COCO^I^t-^OiCO 
i—  (Tt<GOO(MtOO'-iCOO'-H(NOO'—  (OOi—  ( 


J 


COO»Oi—  (ICO*—  I 


tOT-(MT-ocOI>-COT-< 
(MOO        T-H  (M  T-I 


APPENDIX   C 


127 


On  platting  the  values  of  /  and  -  -  as  corresponding  co- 
ordinates, it  becomes  apparent  that  they  are  related  to  each 
other  in  all  cases  as  ordinates  to  a  straight  line;  which  could 
have  been  anticipated  from  the  established  laws  of  fluid 
frictions.  This  is  shown  on  Plate  VI. 


1 

2 

o  g 

J 

a  n*;;  ^3 
5!  -     | 


.32 


e  Q  4 


Values  of  "£" 

From  this  plate  we  get  the  following  three  equations; 
80.0  K  +  2e  +  5u  +  4g  =  18.3, 
•9u          =11.8, 
13  m         =    6.8, 


128  APPENDIX  C 

Va2 

in  which        K~  =  resistance  due  to  one  foot  of  pipe; 


e  —  =  resistance  due  to  one  elbow ; 

in  y-  =  resistance  due  to  one  extra  ferrule  or 
joint  with  ends  reamed; 

u~f~  =  resistance  due  to  one  extra  ferrule  or 
joint  with  ends  unreamed; 

Va2 

g  —  =  resistance  due  to  one  globe  valve. 

So  by  attaching  other  lengths  or  fittings  we  get  other 
equations  and  by  simple  algebra  can  find  the  numerical 
value  of  each  symbol. 

11  2  7  17  2 

Then  KlV-?-  =  c±—     or     c  =  d*K. 

r         d?  r 

Also  the  length  of  pipe  giving  friction  equal  to  that  of 

o 

one  elbow  is  j ,  and  so  with  other  fittings, 
/c 

These  experiments  covered  standard  galvanized  pipes  of 
2,  1J,  1,  |,  and  J  inch  diameter.  With  each  size  pipe,  runs 
were  made  to  find  friction  loss  in  ordinary  elbows,  45°  elbows, 
globe  valves,  return  bends,  unreamed  joints,  and  reamed 
joints.  For  each  combination,  data  was  taken  for  platting 
twelve  to  eighteen  points,  altogether  about  eight  hundred. 
The  results  as  a  whole  are  satisfactory  for  the  2-,  1J-,  and 
1-inch  pipes. 

For  the  f -  and  J-inch  pipes,  especially  the  J-inch  pipe, 
the  results  were  so  irregular,  erratic,  and  conflicting  that  the 
results  finally  recorded  cannot  be  accepted  as  final.  In  the 
light  of  these  results,  it  is  not  probable  that  a  satisfactory 
coefficient  will  ever  be  gotten  for  pipes  under  1  inch;  the 
reason  being  that  in  pipes  of  such  small  diameter,  irregu- 
larities have  relatively  much  greater  effect  than  in  larger 
pipes,  and  the  probability  of  obstructions  lodging  in  such 


APPENDIX  C 


129 


pipes  is  relatively  greater.  In  the  J,-inch  pipe  and  fitting, 
unreamed  joints  were  found  at  which  four-tenths  of  the  area 
was  obstructed,  and  this  with  a  knife  edge.  No  doubt 
consistent  results  could  have  been  gotten  by  using  only 
pipes  that  had  been ' '  plugged  and  reamed/'  and  selected  fill- 
ing, but  these  results  would  ijxrt  have  been  a  safe  guide  for 
practice  unless  such  preparation  of  the  pipe  be  specified. 


s  $  $  5  S  3 

Coefficient  "c" 

The  results  of  these  researches  are  embodied  in  Plate  VII. 
They  show  the  averages  of  such  data  as  seem  worthy  of 
consideration.  The  data  for  pipes  exceeding  2  inches  diam- 


130 


APPENDIX   C 


eter  are  taken  from  the  matter  recorded  in  Appendix  B. 
Verification  of  these  by  the  use  of  the  sensitive  differential 
gauge  is  desirable. 

Table  IX  and  Plates  0  to  IV  of  this  volume  were  worked 
out  with  coefficients  differing  slightly  from  those  here  rec- 
ommended, but  the  errors  are  probably  well  within  those 
ordinarily  effecting  results  in  practice.  Until  the  results  of 
further  research  are  available,  the  author  recommends  the 
use,  in  practice,  of  the  coefficients  taken  from  the  curve 
AB,  Plate  VII. 

In  the  series  of  experiments  referred  to,  the  results  worked 
out  for  the  resistance  of  fittings  were  more  erratic  than  those 
for  straight  pipes.  Hence  no  clain  is  made  for  precision  or 
finality  in  the  results  here  presented.  However,  two  im- 
portant conclusions  are  reached.  One  is  that  the  resistance 
of  globe  valves  has  heretofore  been  underestimated,  and 
the  importance  of  reaming  small  pipe  has  not  been  appre- 
ciated. 


TABLE    OF     LENGTHS    OF    PIPE    IN    FEET    THAT    GIVE 
RESISTANCE  EQUAL  THAT  OF  VARIOUS  FITTINGS 


Diameter 
of  Pipe. 

90°  Elbows. 

Un  reamed 
Joints,  Two 
Ends. 

Reamed  Joints, 
Two  Ends. 

Return 
Bends. 

Globe 

Valves. 

1 

10.0 

2  to  4 

1.0 

10.0 

20.0 

1 

4 

7.0 

a 

1.0 

7.0 

25.0 

5.0 

i  ( 

1.0 

5.0 

40.0 

U 

4.0 

u 

1.0 

4.0 

45.0 

2 

3.5 

(t 

1.0 

3.5 

47.0 

A  series  of  runs  were  made  on  50-foot  lengths  of  rubber- 
lined  armored  hose  such  as  is  used  to  connect  with  com- 
pressed-air tools.  The  scheme  was  the  same  as  that  de- 

v  2 
scribed  for  pipes  and  fittings;  and  the  range  of  -J-  was  the 

same.  The  average  results  are  here  given.  This  includes 
the  resistance  in  a  50-foot  length  with  the  metallic  end 
couplings.  In  these  end  connections  a  considerable  con- 
traction occurs.  For  the  half-inch  hose  the  end  couplings  are 
quarter-inch.  The  excessive  resistance  in  the  half-inch  hose 


APPENDIX   C 


131 


may  have  been  due  to  these  end  contractions  or  to  some 
other  obstruction.  It  is  a  further  illustration  of  the  fact 
that  reliable  coefficients  cannot  be  gotten  for  pipes  of  half- 
inch  diameter  and  less. 


Diameter  of  hose  in  inches 
Resistance  in  50-foot  joints 

I 

950.0^ 

r 

i 

20.0^ 

1 

4.5^ 
r 

H 

2.6^ 
r 

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